Well, I haven't worked with many musicians, but I believe that most (good) drummers help arrange songs, write drum fills, and help with lyrics. Also, some drummers may play other instruments so they can help out with guitar/bass parts.
Now, if your other guitarist and you are the primary songwriters, you should write music on your own time without the drummer there.
Ehh, I don't think so. It's different from band to band. Look at The Who, the majority of their songs were written by Pete Townshend. Whereas The Beatles' songs were majorly written by Lennon and McCartney, with some songs written by Harrison and Ringo. Then look at Queen, they were a complete toss up of each member writing songs.
I don't think every musician is destined for songwriting, but that's fine. You don't want everyone trying to compete with each other all the time. That's when egos start to tear apart bands.