(Some of this is more for those who might be reading to gather ideas for themselves, but the OP might find a few useful things here.)
I've been to Iceland 3 times (from the US). Once pre-MMM-discovery, as part of an overpriced group tour. Once on a week-long backpacking trip through the Westfjords (breathtaking), and once on a 2-week camper van trip around the entire country (unforgettable in all the best ways). Make sure you know what vehicle your camper van will be. Ours was a small SUV. My husband is 6'3". It was rather cramped, but not enough to keep us from getting engaged during the trip. A larger vehicle might actually have been cheaper overall, because we found ourselves going out to eat and/or to share one beer just to get out of the SUV and go inside somewhere on evenings when it was raining. And toward the end of our 11 nights on the road, we decided to treat ourselves to a hotel room, which was about $200, but worth every penny. So keep that in the back of your mind when budgeting. We're both experienced campers, but 11 nights in a cramped SUV started to wear on us.
If camping, take advantage of the fact that even the tiniest towns all have swimming pools. This is code for "yes, we have a swimming pool, but mostly just the kids use it, and the adults go for the numerous hot tubs!" And... in the case of foreigners who are camping... the showers that are included in the nominal entry fee (usually less than $5). You do have to be 100% naked to shower, and a shower is required before entering the pool, but it's no big deal. Men and women have separate showers/locker rooms. Never in my life had I felt so clean while camping! Shower #1, long soak in multiple hot tubs of different temperatures, possibly a dip in the pool, and then shower #2 on the way out!
Definitely get alcohol at the airport, but don't take so long doing it that you miss the (relatively) cheap bus into the city. We arrived at something like 7-8am, so I'm not sure about your 5am arrival time, but I've never seen the place closed. We spent too long looking around, missed the bus, and had to pay an insane amount for a taxi. Never again.
Get out of Reykjavik. Sure, it's a cute town for a day or two, but it's expensive and teeming with tourists. It would be like going to the US and never leaving Manhattan. The Golden Circle is equally expensive and teeming with tourists, and other than Gullfoss, I'm not sure it's really all that magnificent. I know the OP can't do it on this trip, but for others who might be considering a trip, go as far east as you can, and definitely don't miss the Westfjords, either. You can even fly from Reykjavik to Isafjordur, which might be worth doing if you prefer not to rent a car or are short on time. I think it was about $200 or so to fly, and Isafjordur is so tiny you can walk everywhere. There are tour companies that will take you out to do all sorts of adventurous things.
Bonus is good for groceries, and taking whatever you can from home, as someone mentioned, is also smart. Oats and French cheese, for some reason, were cheap when I was there.
I loved all of my trips to Iceland, but I doubt I'll ever go back. All three trips were within the past 10 years, and just in that short timeframe, I noticed a huge influx of tourists and with it, a sensation that the Icelanders are getting awfully tired of tourists. And although there are ways to mitigate the cost, it's still quite expensive. If you've never been, it's absolutely worth going, though! I don't want to discourage anyone!