Each centre will have their own rules about what they accept and what they need clearance for. According to the staying healthy in childcare handbook (which centres use as a reference), a common cold is not excludable as long as the child is well enough to 'handle the day'. Which your son sounds like he is. If they have a nose that constantly runs green, I would say stay home, many parents don't though and it's not too pleasant to have to wipe these noses every 2 seconds, but that comes with the territory. You'd be suprised at the state some parents send their children to care in - good on you for thinking about what's acceptable.
My centre excludes for 24 hours after a vomit, runny poo. We require medical clearance for conjunctivitis and rotavirus and chickenpox.
Oh, and the being sick alot is pretty common for children who haven't been in care before, they will develop a natural immunity after a while.