In the manga, what piece is Chiaki playing at the beginning?
This is the first movement of Beethoven's Piano Concerto no. 1, which is not the concerto being played in the anime. Examining the section where the piano begins to play, we can see a correspondence between the orchestral reduction (as shown in a piano part) and the bars of top line shown in the manga. The lower line corresponds to the opening of the piano part, although a couple notes at the very beginning are missing.

For comparison, here is the relevant section of the piano part. The orchestral part (i.e. the first line) is marked with yellow brackets; the piano (i.e. the second) with blue. (Note that the score Chiaki is using has a different format from the one I found; his includes the orchestral reduction even when the piano is playing, and the reduction occupies a separate line.)