In this excerpt from Bach's BWV 772, does the B flat accidental on the lower staff also apply to the upper staff(second to last beat)?
I know that accidentally usually do not apply to different staffs, but these have same clef. Does the accidental on the upper staff not need to be notated in this case?
As Dom states, each and every stave is a separate entity, and accidentals need to be put in for each changed note. This is in D minor (at least here), and what's happening is the normal melodic minor trick of that era. Melodies going downwards would use a natural minor configuration, while those rising would have a raised 6th and 7th. Thus, Bbs in the left hand, as the tune drops, but the right hand tune rises with a B natural and C# (sharpened earlier in that bar).
In deciding whether an accidental should be printed, one should generally expect that people reading music will be blind to staffs that don't contain their part(s), and may be blind to music which they are not performing even when it shares a staff with music they are; one should add cautionary accidentals if necessary to ensure that such possible blindness will not affect the pitches to be performed.
In your example, a keyboard piece in C major has two treble clefs, and the left hand (lower staff) plays a mid-staff B♭ near the start of a measure, then if the right hand plays that same staff note later in the measure it should be explicitly marked with either ♭ or (♮). Indeed, such a marking should be used in cases where the staffs used different clefs (e.g. if the left hand, notated with bass-clef, played the F♯ above middle C, and then the right hand, notated with treble clef, played its first-space F.
Note also that if e.g. in C major one performer performs a B♭ in the middle of treble clef, and another performer simultaneously performs a B♮ in the middle of treble clef, it may be helpful to mark the latter note with a cautionary accidental (♮). Such a marking may not be necessary from a performance standard, but in its absence the dissonance might make performers wonder if the "B♮" might be actually be a misprinted B♭. In cases where confusion is unlikely, no accidental is required, but when confusion would otherwise be likely, a cautionary accidental can help.
Returning to your example, the B♭ shouldn't apply to the right hand, but I think that notationally it would have been appropriate to mark the right hand B with a (♮).
No they do not. If you wanted to represent the accidental on a different staff you would need to write it in. Each staff is independent of the other even if they are linked as in your example.
No, accidentals are always staff independent.
The reason you have a Bb in at the beginning of the bar and a B natural at the end is because the Bb is pulling down to the A and the B natural is leading up to the C.
