Hi,
We've rented a car in Cancun for the last 5 or 6 years. We stay about 30 minutes south of Cancun, so that's part of the reason. I'll try and answer your questions as best as possible.
* Around town is there plenty of free parking?
Well, I haven't spent more than one or two nights in Cancun, so can't really comment on the parking there. I've never had a problem with parking, however.
* Is it safe to drive to Chichin Itza and are there free places to park when you get there?
We did Chichen Itza a couple of years ago. We took the toll road there, and from Cancun, I think it was maybe 1 1/2 hours???? Can't really remember. I don't remember the tolls being that exorbitant, however. There is parking available — can't remember if it was free, but couldn't have been too expensive.
We loved touring the ruins!! I would highly recommend visiting. We were fortunate enough to have visited when you could still climb the one pyramid; I understand they've closed it off now. If you go, I'd suggest hiring a guide — our guide could have taught a college level class in Mayan history, and we got so much more out of our tour with all his knowledge.
We decided to take the local roads on the way back, just for a change of pace. It was definitely slower, but we really enjoyed it. You drive through a number of towns, and it was interesting to see how the culture really changes when you get away from the big, touristy cities. The younger girls were all in skirts and white blouses, and there was definitely an 'old school' feel to everything. We stopped in Valladolid, walked around the town a bit, and had a wonderful lunch at a Mexican/Mayan restaurant. (I can try and look it up if you're interested)
* Can you take your car to "Isla Mujeres" or is the ferry just for people?
The ferries are only car only, I believe, but you really don't need a car on the island anyway. It's pretty small, and we've always just walked around. You can always grab a taxi if you need to get somewhere. Isla Mujeres isn't our favorite place to visit, which is why we've only been there a couple of times.
* What about driving to the ruins in Tulum? Is it safe for two non spanish speaking tourists?
They've been constantly improving Highway 305, and it's now a multi-lane, divided road all the way to Tulum. I never felt it was a dangerous drive, even when both directions shared the middle lane to pass.

And as far as not speaking the language, it's not an issue at all. The Yucatan Peninsula bases a large part of their economy on tourism; thus, you'll find English spoken most everywhere.
!HELP! Any information/suggestions would be appreciated
My family and I have driven throughout Cancun/Playa del Carmen/Tulum, etc. for the last six years, and I've never been pulled over by the police. As long as you aren't standing out from the flow of traffic, or doing something stupid, you'll be fine. I would agree that if you do get pulled over, you simply ask the officer if you can 'pay the fine directly to him'?
It could just be my observation, but I've found that while people may drive a bit more aggressively in Mexico, they also pay
a lot more attention to the surrounding traffic. You don't see people eating their lunch, talking on their cel phone, putting on makeup, etc. while driving. In some ways, I almost prefer driving down there — at least everyone's paying attention.
I wouldn't think of visiting the Mayan Riviera without a rental car. But that's just one person's opinion.
Best of luck!
Tom