Yes you can do the FAA exam here in Europe but you are better off doing them in the US at half the price.
If you want to use the priviliges of your licence to the fullest you would need to have two licences
Yes just do a search on google for FAA schools in your area , there are not many but there are a few.
Florida summer can be severe and you could loose a few days flying due to storms . If you are not used to the heat and dont mind staying a few days longer then there shold not be a problem.
Just stay away from a schools called OBA or they are now cahnging there name to EASA