Don't know, never used the DS1307, can you send a linkt to a datasheet? The drawing looks very nice.
I believe you have the right arduino pins (or at worst they are swapped).
For push buttons, why not use the internal pullups and switch to ground instead? Save on a few parts.
Which pins are PC0 & PC5? Aren't those the analog pins?
You already have A5 committed to SCL above, so that wouldn't be a good choice for an input pin.