This is their website
I don't fully understand your requirements, but if you have a look at the website I linked to, it will show you what they offer.
I can fully recommend them (I did a FAA instrument, commercial, multi-engine, instructor certificates and ratings there) as can many others.
You have an FAA Private Certificate, correct?
To answer your numerical points;
1, Yes, but I don't know if you already have a standalone FAA private, or a UK one, or what?!
2, Yes, they are a mom & pop outfit, read their website.
3, They are SEVIS approved for the M-1 visa, and offer part 61 and part 141 training.
4, I don't know what medical you are going on about, FAA or JAA. To do the FAA stuff you will need only a FAA Second Class medical. No need for any JAA one, unless you have a based-on (piggyback) FAA private, in which case your JAA PPL must be valid medical-wise... I don't think the FAA care if you have a lapsed SEP rating on your JAA licence, as you can just do a flight review to get your piggyback one legal and use that.
Once you have an FAA Commercial, you can forget about your private... But you will likely do the Instrument Rating first, so you need a valid licence/certificate in order to complete that
5, They are in Arizona... The same state where Oxford and Lufthansa train. It's a desert environment, and flyable pretty much 99.9% of the year. Forget Florida.
6, I'm not biased, just recommending a school. I'd never heard of them until someone told me, so I'm just passing on the info.