It really depends on your parish and the priest doing the baptism.

I'm not Catholic so it didn't matter, our wedding wasn't in the catholic church (however we did have a marriage blessing at the baptism as it was something we wanted to do for MIL, as it didn't mean much to us but a lot to her). I have friends who were never married but kids were baptised fine, but some churches don't allow it, so you just have to see what the church you have in mind is ok with. Godparents are fine as long as you have 1 Catholic godparent. We couldn't have 2 of our godparents present, so we had stand ins. We gave I think $250 for the baptism, and we had a family priest come to our parish to perform the ceremony but he wouldn't take a donation so we just made him stay for lunch We were never interviewed but I think it was because we had our own priest. Yes you can speficty when the ceremony is held providing the priest is available. We had ours done post Mass. It was a private ceremony.