Yes, my wife is half Icelandic and I have been to Iceland at least 25 times to visit my mother-in-law and her large extended family, most recently a couple of weeks ago for the holidays. I have successfully rented C172's and Diamond DA40 NG's at least a half dozen of those trips. It is a really cool country to fly, and they have always taken my US credentials. Mainland Europe would require a lot of extra effort to get EASA certified, especially if you want to fly IFR. But as you said, very weather dependent. Conditions can change quickly unless there is a large high pressure system over the north Atlantic (by far the best time to fly). All of my flights and rentals have been from flight schools based at BIRK, the Reykjavik city airport. I made arrangements to fly with a safety pilot in a DA42 last August but weather conditions did not cooperate. Over the years, I have flown the Golden Circle, out over central Iceland and Hekkla volcano, plus up over the northwest fjords (by far the best flight) and Snaefellsness peninsula.
You definitely need a check ride and it's good to download and study the Iceland AIP from the web. They have some funky VFR procedures where you must fly specific departure and arrival routes and waypoints in their Class D, somewhat similar to very simple DPs and STARs in the US.
A couple of years ago I got to fly over the active Reykjanes Peninsula volcano near the international airport BIKF. A link to my Youtube (sorry for poor quality, alot of it is shot handheld by my copilot as I did not bring all my good cameras from the states). Let me know if you want more specifics on flying there.
Iceland Volcano Video