Puerto Rico: This is quite a gem of a destination. Aside from being beautiful and interesting and always warm, no passport req, no currency or credit card issues (don't need to notify banks about leaving the country-- you're not, also no need to worry about currency planning, just take home the $$ you don't spend), your car insurance is still in effect, the whole country is bilingual, there is a thriving middle class so there are great restaurants, shops, supermarkets already in place, no feeling of a wall separating you economically or racially from the local population, and no crime to speak of. If you don't want to get sunburned, wear a hat and sunscreen and stay out of the midday sun, and the equator is not particularly close, over a 1000 miles south in Brazil.
Anyway, 4 nights in old San Juan is probably too long, there is no beach there and it's pretty small, and the cruise ships dock right there and fill the place up every day or two. Far better to stay in Condado, a beachy, pedestrian, urban area just east of old San Juan. There is an easy public bus for 75 cents or $10 for a taxi each way that will take you to old San Juan. Both the forts are very interesting, San Felipe near the harbor entrance is more popular but a little smaller than San Cristobal, which is east and near the bus station and docks.
If you want to do a biobay swim, very recommended, we really liked Aleli tours in La Parguera, note that this is on the opposite side of the island and a 2 hour drive, but easy, almost all freeway then nice 2 lane highway. Driving is pretty straightforward in Puerto Rico but it is a bit crazy like Miami: a mix of very fast and very slow, wandering drivers, but mostly slow and wandering. The driving warnings in guidebooks refer only to old San Juan, not to Condado or Puerto Rico in general. The freeways come right to both Condado and old San Juan.
I can't help you on Key West, however there is a nonstop flight between San Juan and Ft. Myers if the ferry interests you.
I remember carefully planning my visit to the Everglades to avoid driving the no cash stretches of toll roads in Miami, so yes they are there. The same problem exists in Puerto Rico but all rental cars come equipped for it.
Mosquitoes: there aren't any in the winter so I have no experience. April may be too early for them, they are a summer problem.
Just a suggestion but there is plenty to see in PR to fill up your 6 days.