When I startup windows, ClamWin also starts. This seems silly to me as it doesn't do 'on the fly'-scanning. How do I stop ClamWin from doing this?

I have 'solved' this by using CCLeaner which allows me to disable services started at boot. I am still wondering whether there is a more simple way to achieve this...

you can set startup items with msconfig

clamwin tray however is an essential component and performs scheduled virus database updates and scheduled scans

I was not aware of 'msconfig'. I'd expected something like that to be found in the control panel. I either missed it, or it isn't there (but now that I know of its existence, it doesn't really matter ).

I do have one additional question. I don't use scheduled scans and I am perfectly willing to press "update database" myself. Would I be missing anything when disabling the 'tray' in this case?

Nope. If you don't use scheduled scans/auto updates, you can safely forget the tray component.