At the very least, during the summer you will need a good 4/3. You will need boots and gloves. I highly recommend a hood also. While it is possible to surf without a hood in the summer, the weather is usually windy and cloudy and wet. So, in addition to dealing with the cold water, you will have to deal with cold, clammy weather as well. I have surfed as long as an hour and a half on the warmest day of the summer without a hood. But I was shaking with cold by the end of the session! Water temps in summer average mid 40's. Air temps average around 50.
In the winter, I recomment the very warmest gear you can find. Water temps average mid 30's, Air temps average 30 and below, with strong winds causing teens and below windchills.
2. Is there anywhere I can rent a board, wetsuit, etc.?
As far as I know, there are NO surfshops in Iceland. A fellow American talked to a local Icelandic surfer, who said that there was once a surfshop in Reykjavik, but it is no longer open. There is a shop that sells surf clothing, however. Right now, there is nowhere to rent gear that I am aware of. I have an extra board that I may be able to rent to you. Contact me if you need more info.
3. Are there any local surfers? How often do you surf alone? Etc...
Yes, there are some local Icelandic surfers. Most of the time, I have seen them at Thorlakshofn, although I have seen surfers at a few of the other breaks also. There are one or two American surfers who live out here also. However, I have only surfed with other people a few times, mostly if I found someone who was visiting and contacted me prior to coming up here. I would estimate something like 6 sessions out of 100+ in the last year have been with other people.
The locals I have met, surfer and non surfer, are very polite and helpful. Illiteracy and crime are virtually non-existent in Iceland, and every Icelander I've met speaks very good English. There are no crowds and no localism. If you come here, be polite and friendly and you shouldn't have any problems at all.
4. How much surf do you get? What kind of board do I need?
In the winter, so far, the surf has been huge for weeks on end. Double overhead plus is the norm with strong, frequently extreme 50+ knot winds. During spring I averaged 3-5 sessions a week, and summer, it wasn't usually flat more than 2 weeks or so. Fall, I averaged about 3-4 sessions a week. January, I had about 5 sessions, and I'm looking at maybe 3 sessions for February.
For boards, if you're a charger, bring a big, thick gun. There is some seriously big surf here. If you're not a charger, a regular shortboard of fish will work fine. I've gotten the most use out of my 6'8 pintail.