Hello!
...cause youŽd leave through arrivals and go in through another building.
Not any more (and I have been there very recently). There is a staff/GA entrance and exit to the left of the terminal building where you still have to call Delta Aerotaxi by telephone. They will be there to pick you up by the time it takes you and/or your passengers to get through security. Airside, they have a crew lounge and selfbriefing facilities and will provide coffee, hot water and ice if required. I do not know about catering, but I am pretty sure they can arrange that.
Taxis have been no problem, there were dozens of them waiting in front of the terminal. The fare to the town centre is now fixed at 25 Euros, no need to negotiate any longer.
Florence is PPR for parking due to very limited space - be sure to get your permission well in advance, especially for overnight parking. During times of high traffic like trade fairs you will normally get no parking permission and will have to position your aeroplane to Siena or Pisa after disembarking the passengers.
Regarding hotels I always find it very difficult to give advice, because every person has his/her own preferences. Last time we were put up in the "Home Hotel" (google should know about it) and I found it quite pleasant. Close to the river Arno and within ten minutes walking distance to the town centre. Nice breakfast buffet (not just a cup of Espresso and a brioche like in many other hotels in Italy), free wifi, free minibar (not much inside, though) and working aircondition (essential at 39 deg. C of last weekend). Negotiate a crew rate through your company in advance, otherwise you will pay an awful lot more.
I have been to Florence many times and when I can choose myself, I will always opt for hiring a car and driving half an hour up the mountains and go to some small country hotel there. Much cooler climate, not the crowds of holidaymakers pushing you through the baking hot, dusty and stinking streets (there are cars and mopeds everywhere all the time). And better food and wine at half the price.
Enjoy your stay!