- Whether to stay in San Juan the whole time or move to a hotel somewhere else (the rainforest)?
You say you want to split your 5 days between the beach and exploring. 5 days is not a lot of time, so you should stay in an area where you can do both within relatively close distances. Since this will be your first time in Puerto Rico, I would suggest you stay in a San Juan area beachfront hotel for the first 2-3 nights and do your beach thing there. That also allows you to explore the old San Juan forts, and walk the old cobblestone streets to see historic buildings, do some shopping, etc.
For the next 2-3 nights you might consider staying on the east end of the island, where you can see the rain forest and do some trips to offshore islands.
- Good but affordable hotel or B&B (the Marriotts are close to $200/night with all their ridiculous fees)?
Puerto Rico is not a cheap island to visit, but you can do much better than $200 per night. Since DW and I are "locals" during the winter in our condo, we're not the best to offer hotel advice. But we've heard good things about the Ceiba Country Inn B&B on the east side, not sure of the prices. For the San Juan area, the Caribe Hilton is right on the beach and close to old San Juan, but probably pricey.
- Worth it to get a rental car or not?
Yes it is, but you have to choose when to use it. I would not drive into old San Juan, try to get a tour from your hotel, or hire a cab by the hour if not too expensive. Not that you can't drive in there but be prepared for traffic and confusing streets. It's a 500 year old city, and the traffic patterns aren't exactly up to date. You should use the car to drive to the east coast, the roads are modern and you won't have any problems.
- Worth it to take el cheapo ferry to one of the small islands? Can you do that as a day trip or would you have to lodge there?
Remember, you only have 5 days
Yes, if you have a day, you can do a day trip on the $3 ferry to either Culebra or Vieques. With limited time, I would take the morning ferry from Fajardo to Culebra that leaves around 9AM (but check the time) and returns around 4PM. At Culebra you take a taxi to Flamenco beach (the taxis are all there waiting) which was named as one the top ten beaches in the world by the travel channel. The taxis bring you back to the ferry and you will be back in your hotel that evening.
"The biobay sounds awesome. Radiotelescope at Arecibo might be worth a stop. Other good things to do/see? Any photos you want to share are appreciated."
The best biobay is in Vieques, but there is one in Fajardo. However, forget it, unless you will be there on a night with no moon. You will be disappointed if there is any moonlight, because you can't see the phosphorescence. In case you are there during a new moon, the Vieques bay is one of the most impressive natural wonders I've seen anywhere in the world. But you have to stay overnight to see it. If you want to do the Fajardo one, there are kayak tours that leave every night, and your hotel will have info. If you do stay at the Ceiba Inn or any other hotel on the east coast, you will only be 20 minutes from either the cheapo ferries or the Fajardo biobay. If you have a couple of hours to spare, go into El Conquistador hotel for the view, it's just fantastic, way up on a high bluff, and on a clear day you can see all the way to St Thomas. There is also a guided tour of the Fajardo lighthouse that takes you on a small trolley car through mangroves, hills, and beaches, and they tell you all about the history and ecology of the area. It's run by a nonprofit land trust, and you need reservations. They do an english language tour. But, that may be a lot in 5 days.
The radiotelescope is in the opposite direction from San Juan and is an all day trip, so you can put that off to another time.
Let me know if you have specific questions, and PM me if you like.