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 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.