Last July, I backpacked in Iceland for a week and then rented a small RV and drove around for another 3 weeks.  October is getting pretty late in the season, so plan for snow.  I would highly recommend bringing a GPS with some of the GPX files for the trails that you want to go on as the trail markers may not be visible/existent.  Your best bet may be the southern section of the Ring Road: 1) I would highly recommend hiking Skogafoss as the falls are easy to get to AND when you continue up the trail, there are multiple, multiple more falls above Skogafoss 2) Jökulsárlón Glacier Lagoon is a beautiful spot.  You cannot camp there but there are places to camp just west (or east of there). 3) Hofn – just for the lobster (if it is still in season).  We had an incredible meal in Hofn.
There is always the Golden Circle around Reykjavik. Â I enjoyed Myvatin but it may be pushing it for a 1 week trip. Â The highlight for me was the Teutonic rift and the site of the first parliament. Enjoy your trip.
Forget the hot dogs and enjoy the Skyr, best lamb in the world and great seafood.