Does a key signature stay the same after a clef change? I have provided an example; it is in the second staff of the lower system as indicated by the blue arrow.
Yes the key signature remains the same. Piano music may have several instances where both hands play high and really there is no reason to reiterate the key signature when the change of clef is just to not use unruly ledger lines.
Yes it stays in the same key. There is a very specific way to notate a key change on sheet music and in this case if it were changing to the key of C major/A minor you would see all the places there would be flats have naturals in their place. You can even see in the chord symbol that the D notes are still flat in that measure.
For a temporary change of clef, there is no need. However, if the 'left hand' continues to use the bass clef , say, in the next line, it will revert to the proper key sig., with the four flats (in this case) in the appropriate places for the bass clef, which obviously will be signed.