I think that you have time to go to Thingvellir and Gulfoss and drive to your accommodation in Hellissandur in one day. Those sights are worth the time spent. I would say we spent about an hour and a half at Thingvellir, not counting picking up some sandwiches in the visitors center and eating them. At Gulfoss, we spent about 45 minutes at the upper falls, but didn't go to the lower falls. Wear rainjackets--you will get wet from the heavy mist from the falls!
Backing up a bit, you have to allow time to pick up your car and drive to Thingvellir. Google maps shows about 3 hours driving time to Hellissandur from Thingvellir. After that, I don't know how much time you would have exploring the peninsular, but with pretty long days you should have some time. The key would be an early start, as you said.
We stayed east of the peninsula and drove there as a day trip. Most of the sights seem to be along the south coast, out towards the end where the national park is and drove that first. We wanted to make it as far as Stykkisholmur for dinner, but skipped it so we weren't driving back in the dark. It's supposed to be a nice fishing village on the north coast.
At many of the sights such as Thingvellir, you have to pay for parking. You just drive in and find a spot, then there are machines inside the visitors center. You have to enter your license plate number and pay. Sometimes you get a ticket to put in your windshield, but at some places like Thingvellir they take a picture of your license as you drive in. At the end of the day if the ticket machine doesn't show a record of that license having paid, they send a ticket to the rental company who sends it to you. It seemed that if you didn't have to pay for parking, then you had to pay for the toilets, usually 1 ISK. That was generally on the honor system, with bucket outside the doors. We saw $1 bills, 1 euro coins, as well as ISK in there.
Good luck and enjoy your whole trip!
