Those are a lot of excellent questions. I hope that I can give some suggestions that might help.
As others have said, people do learn differently so some practice methods may be more effective for a particular person than others, which might work better for someone else. But there are a few things that I have found can be helpful across the board.
1. When you start a new piece, try reading through at a reasonable tempo just to get an idea of the shape and layout of the piece. Also, it never hurts to listen to different recordings of the piece if possible, both with and without the score.
2. Next, find and mark the strong cadence points in the piece. These will indicate the major sections of the piece. These will generally be the (perfect) authentic cadences. Each period -- particularly the Baroque and Classical periods -- have a rather specific vocabulary for these cadences and as you learn this vocabulary, these will begin to be almost second nature. Note that particularly in the Baroque, there may be some overlap between the cadence and the beginning of the next phrase.
Along with finding these cadence points, mark the key of the cadence. This can be particularly helpful if your goal is to memorize.
3. Now, play through and mark all of the places where you have particular technical challenges. Include a few notes at the beginning and end so that they have context with what preceeds and follows. I call these "fractures" (from "Playing the Piano for Pleasure" by Charles Cooke), because these are the spots that we are setting like a broken bone. When they heal, they become the strongest spots. These will be the parts that you will want to concentrate on first. This way, you will not spend your time starting at the beginning and playing until it falls apart and then start to work there. Also, if you are aware of the particular problem spots, you may discover that two or more are similar and therefore work on one will go a long way towards solving another. How to practice each of these spots will depend on the particular problem, which I'll talk about later.
As you fix these fractures, then put them in place with the parts that surround it. Once in place, start work on the next fracture.
4. Set an initial goal tempo. This need not be the desired performance tempo, but it should be fast enough that as you put everything together, the piece is identifiable.
Now for some specific thoughts on practice techniques.
1. It is particularly important to NOT "correct" as you play -- i.e. if you play a wrong note, don't just change it to the correct note and continue. You have not only not corrected anything, you have made the error bigger! You have altered the rhythm, the melody, the harmony, and the technique and this is what you have actually practiced (as opposed to "un-practicing" the mistake.)
If you do make an error, stop and analyze the error. It should be noted that the audible error may not be the source of the problem but rather a symptom of something that happened at an earlier point.
If it is a note error, what note did you play and what note was intended? How close was the note that you played to the note intended? Say the note played was "F" and the intended note was "F#", could be a read error due to key signature or an accidental earlier in the measure. Did you use the intended finger? If not, then it may be a technical issue -- you are unsure of the fingering there, or you missed the fingering at an earlier point that wasn't as noticeable and in the process, this was the point that you were no longer able to compensate. Or maybe the passage has an awkward fingering.
2. Try to avoid read errors from the beginning. The major causes of read errors involve the key signatures and accidentals.
Key signature read errors primarily occur by forgetting it. I've been having my students put a box around each note that is affected by the key signature when they first start a piece so that they are aware from the beginning that they need to do something on these notes. Not all read errors sound bad.
Another key signature problem occurs when it changes in the course of the piece. This is similar to read errors due to accidentals (particularly when an accidental is cancelled by the bar-line). If the new section has cancelled the flats or sharps of the old key signature, write in the cancellation for the first few occurances of that note. Do this with accidentals also to indicate the cancellation after the bar line. And if necessary, write in the accidental by the other affected notes of the measure. If sharps or flats have been added, using the box method above can help, since these are now parts of the key signature.
Another read error occurs when clefs change. Make sure that you are well aware of that and not continue reading as if you were the other clef.
3. Fingering -- Be consistent. When working out the fingering, people often start at the beginning of a passage and work from there. Rather than that, first find the particular points the will require a certain finger. These points are the end of a passage so that you have the necessary finger to continue into the next, and the highest and lowest points in the passage. In the R.H., you do not generally want to end up with fingers 1 or 2 on the highest note (L.H. lowest note), nor do you want to end up on 4 or 5 on the lowest note of a R.H. passage (highest in the L.H.) particularly if it is going to immediately go back the other way. Once you know those points, you can work out the rest.
Clearly mark each place that involves a finger crossing, a finger substitution, or a change of fingering on a repeated note, crossings involving the upper fingers (4 over 5). Also, clearly mark non-standard fingering for such things as scales and arpeggios. For example, a passage may be a C major scale, but it may continue to the octave D. The standard fingering would cross to 1 on the high C and place 2 on the D. If it were then to return, you would have another crossing (1 - 4 on C to B) almost immediately. In this instance an non-standard fingering such as 1234 1234 may be more effective.
Consistent fingering will be particularly important when working with contrapuntal music with 3 or more voices. Nothing will get you in trouble faster in this music than not being clear on the fingering.
4. Hands separate practice is always a tool, whether it works for you doing the entire piece that way or simply particular sections. I generally reserve it for particular problematic spots rather than working through the entire piece that way. When using hands separate, be particularly careful of notes that may be part of a middle voice that may be divided between the hands -- hold them for the correct duration and with the correct volume for its part.
Related to Hands Separate practice is "Parts Separate" practice. This is particularly useful in fugues. In this, you practice each voice separately with the intended fingering. This is helpful for those middle voices that are divided between hands. It allows you to listen to the part and work on consistency of tone through the line. For me, it takes a little prep. I write the fugue in open score and end up writing in about every finger so that they will be correct when the other voice is added. If I am being particularly OCD, I then practice two parts together, with correct fingering, such as S T, S B, A T, A B, etc.
5. Slow practice is a valuable tool, BUT it can also create problems. For me, slow practice works best to work out the initial fingering of a passage, particularly if it is "non-standard", to coordinate hands in fugues or intricate passages, and to work out complex rhythms. But my OCD turns into ADD if I turn an invention that lasts 1:20 into something that lasts as long as a Mahler Symphony. Therefore I generally reserve it for my fractures or for short sections until I work them up to my first goal tempo.
When I do do slow practice, I use a metronome (actually the groove box on my keyboard since it is more fun to have a drum rhythm than a "Click Click Click"), start at a tempo that I can play correctly, and gradually increase the tempo by one or two bpm. Doing this, I can generally only hold my attention through an increase of about 10 - 15 bpm. The next time, I will usually start a little faster. When my attention begins to wane, I might "accidently" push the tempo up by 3 - 5 bpm. In my case, I rarely make it through this way to my goal, but after I have let the section rest a day or two, I find that I can work it up at a less annoying rate of increase.
The problem with slow practice is that it can lead to over exaggerating the wrong technique! Slow practice is good for developing technique that requires a lot of finger motion. It is also okay to exaggerate an intended motion to help develop muscle memory (such as lifting fingers higher than will be used in the end), but if the passage is going to require less fingery motions and more arm based motions with the finger closer to the keys, slow practice can actually get in the way.
Some motions are not practical below a certain rate. Motions that use gravity, incorporate a sense of rebound, or require momentum require a certain minimum speed to work. Also, sometimes, fingerings that work at slower tempi will be less effective at faster ones. It is possible to reach a tempo "Wall" where you won't get any faster without changing the technique, which means that all of that work up to that point was for naught in this particular piece.
Three practice techniques that are useful and allow you to practice fast slowly (or slow quickly) are blocking, rhythms, and "backward practice."
Blocking is grouping notes together that fall within a hand position. Of course arpeggiated passages are a prime candidate. Broken chords that do not involve finger crossings, such as in the Prelude #1 of the WTC Bk. I, simply need to be blocked as chord progressions with the desired fingering.
Arpeggiations that do involve finger crossings can be blocked with the thumb note then the notes before the crossing, then the next thumb. You can do this in an even rhythm or by holding the notes for the appropriate combined length of time for the notes that are blocked (i.e. if two notes blocked together are an eight note each, you would hold them for a quarter.) This can also be done with scales passages.
Rhythms can combine practicing both slow and fast at the same time. Consider a 16th note fast scale passage. Dotted rhythms can allow us to work at half the speed with half the notes played at tempo. If we were to play the passage as dotted 16th, 32nd at half speed, the second note is a sixteenth at full speed. Then you reverse the rhythm (32nd, dotted 16th). Then you can play the notes up to and including the first crossing (such as CDE F -- 1 2 3 1) using 32nds or at tempo sixteenths to the F and stopping on the F. Then to get past the crossing, get the 2312 (DEFG) stopping on the G etc. (its easier to show than to describe).
"Backward Practice" is not playing the piece in retrograde. It is actually a type of "add a note" practice but starting at the end of the passage and working back towards the beginning by adding a note, beat, measure or whatever is the convenient unit. The advantages to this rather than starting at the beginning and doing "add a note" are first, you have a definite stopping point and are less likely to just continue, and second, you know the fingering and hand position that you need to end on. By working backwards, you will be able to spot more quickly if you need to rework any fingering leading up to this point before you have spent hours practicing one only to find that you need to change it to get to where you need to be. Again, I generally reserve this for sections that need particular work rather than the whole piece.
A couple of final thoughts. Every practice session on a particular piece reaches a point of diminishing returns. You get to the point where nothing new is happening, or worse, you start doing the passage worse than you did earlier. This is the time to stop and go to something else. Your mind and your muscles need some time to digest what they have learned. The next time you come back to the piece, things will generally come together more quickly and better.
On a similar vein, we often reach a plateau in a piece. We get to close to it and have several sessions where nothing seems to happen. At these times, the best practice is (heaven forbid I should say this) to not practice that piece for a few days (sometimes even longer). Maybe even include activities at your piano or organ that are substantially different. Practice sight reading, play through pieces you have learned, work on some keyboard theory and harmony. When you come back refreshed you will usually be surprised at how well things happen. Just as above, your mind and muscles need time to put it all together. Also, very often, the solution to a particular problem my occur in another piece.
I hope that this gives you some ideas that will help.
"I never do anything always." - Pablo Casals