Will an iPhone bought in the US work in Spain?

It depends on the carrier, but most likely. That said, the warranty has limitations that may not be valid in Spain. Ideally, you should purchase a device in the country where it will primarily be used.

Yes, just make sure that it is unlocked from the carrier you purchased it from. And it would probably be best to contact the carrier you wish to use in Spain and have them verify your IMEI number.