From a scenery perspective, I would recommend riding A1A vs US 1 in Florida. Both will have its stretches of strip malls but both will have stretches of nothing, however A1A will have beaches/ocean vs the Indian River or pine trees for your viewing pleasure. A1A will be a lower posted speed limit as well along with less commercial traffic.
I believe since they've redone US 1 from Florida City to Key Largo they have expanded the shoulder to allow for "safer" bike travel, but it can still be very dicey. There is always a lot of traffic on US1 on that stretch regardless of time of day. An alternative is Card Sound Rd. Unfortunately, it is 2 lanes without a shoulder, but there is less traffic since it is a toll road.

Once you get into Key Largo, you should be good to go. There is a dedicated bike lane/path from Largo to Key West and I believe most of it is off the shoulder (physical separation from the road and bike path).

Regardless of when you go, it will be hot/warm and humid. It doesn't cool down here until mid November. We have a freak cold front once in a while in October, but for the most part will remain in the mid to upper 80s. You will constantly battle the trade winds in the keys. It keeps the temps a few degrees cooler, but they can be a byatch to ride in.