Driving in PR is as safe as any major city in the U.S. - so yes, it can get a little crazy at times, but no worse than L.A. or Miami or D.C. Once you get out of San Juan it's easy. If you get onto any secondary roads, they can have some potholes, if you stay on the main roads, it shouldn't be a problem.
It's an easy drive out to El Yunque, maybe an hour and 20 minutes. Mid day, the traffic should not be bad and coming back most people will be leaving San Juan. Leaving at 10:00 pm should give you more than enough cushion.
I only rented from the Airport, so I can't help you there.
It's easily doable, if you can rent a car. Good Luck!