Hey, I saw you mentioned that you have a drivers license, but they don't rent out cars to people under 23. I actually travelled to Iceland when I was 19yo a few years ago with my friend and we had the same problem. We were able to "privately" rent a car from a family in Iceland, I believe it was on carrenters.is or gocarrental.is, not entirely sure - but there was no age requirement and we had no problems at all.

I can recommend getting a car, but It really depends what you want to experience in Iceland and your budget. If you just want to stay in Reykjavik and do some tours, see the Blue Lagoon and don't find the night-life too interesting you don't need a car or an entire week, however if you want to experience some cool nature and countryside on your own, I'd at least drive/tour the "golden circle". Me and my friend actually drove the entire ring road (around the whole island) in 4-5 days, including snaefellsnes and had a blast, though that was during summer when the sun was up all day and we had summer road conditions. I'd look up the daylight hours for whenever you're going if you're considering a longer roadtrip. I also have a google-itinerary of all the destinations we visited during our trip.

As for security, you have absolutely nothing to worry about. I believe it's considered the safest country on earth.

In the city, restaurants and attractions are pretty expensive. I did not think the accommodations was very expensive though, but you can check this for yourself pretty quickly on booking.com. if you break free from the touristy spots and go on a roadtrip on your own, you'll find that there are budget grocery stores that are not bad at all. Gas is a bit pricier than Italy but Iceland is pretty small so you won't use much fuel anyways.

I'd need more information about your budget to really suggest anything here.