I will answer your questions from what I know of my area. Your area may have different laws, policies, culture, etc.
1. do many infants get placed into foster care?
Yes, but the vast majority reunite with their biological mothers or other relatives. There is actually a very high need for foster homes for infants under 6 weeks because those babies cannot go to daycare.
2. Can I specify, if we become foster parents, that we only want a baby?
3. Can we say no if they offer us a child outside our specifications, and will it reflect poorly on us if we did decline a placement?
Yes you can say no to any placement.
In my experience, we only stopped getting phone calls for new placements after two years when we made it clear we were only interested in fostering children biologically related to the one we already had. For two years we received call after call after call that we just ignored or said NO to.
4. do they try to place children in homes of the same race - we are open to any and all races/nationalities.
Depends on the worker. In our situation, we had a placement worker that didn't care about race but then we had a caseworker who appeared to have some racial biases and tried to remove our FS and place him in various other same-race homes.
If you really want to adopt a baby and you think you'd have a hard time dealing with the uncertainty of fostercare then you should do a private domestic adoption instead. Find an agency or an individual to help you. But if you're VERY flexible and you're cabable of handling a lot of stress, incompetence, and uncertainty OR you simply cannot afford a private adoption, then opt for a fostercare adoption. Personally, we will NOT be doing another fostercare adoption, at least not any time soon. We DO plan to adopt another child and we're pretty flexible about age and race and we prefer an open adoption, but there's simply no way we will abuse ourselves by interacting with the fostercare system again. I say that as someone strongly commited to adoption, children, and family. The system is broken.