Yes, 3 sharps is the key signature for A Major
The second line changes Time Signature. The "C" is an abbreviation for 4/4 time, also known as Common time, now having 4 counts in the measure and 4 counts in a whole note. The first line has the time signature written out as 3/4 time, with 3 counts in each measure.
Key signatures are independent of instruments and are the same in all cases, so yes, the same as the keyboard notes.
Having no sharps puts you in A minor/ C major (all letter notes/white keys), and the key moves up a 5th every time a sharp is added.
When the first sharp is added you have moved up to G major. The second sharp moves you to D major, and the third sharp puts you in A major.
The Suzuki books have Piano accompaniment for the exercises. The key signature is added to the exercise even though you aren't playing anything specific to A major so it matches what the piano is playing.