Driving in Iceland: a good idea?

Below are the answers to your questions, respectively:

1) Reykjavík is a pretty nice and relaxing city. How much time you want to spend there, depends on what you like and what you want to do there. If you like going to the pubs and just relax for sometime, then I would recommend to stay for a couple of days in Reykjavík.

2) If you want to take the Route 1 (Ring Road), 3 or 4 days would be good enough to drive around and look at some attractions like (Snæfellsnes - mentioned in 'Journey to the center of Earth' novel/movie, Westfjords, Akureyri - Northern Iceland, Lake Myvatn, Vatnajökull, Skogarfoss, etc). You can also do it in shorter time. If it was me, I would take more time and explore all the places, stay overnight at a couple of places where I can relax and visit some attractions around, etc. Apart from this Route 1 (Ring Road), you can drive around the Golden circle, which has some of the main attractions of Iceland (Geysir, Þingvellir, Gulfoss waterfall, Central Iceland, etc). Central Iceland is really exotic and it feels like you are on a different planet or moon. Some of the US astronauts (including Neil Armstrong) are said to have been in this part of Iceland, as part of a training program, before the Moon expedition.

3 & 4) With a normal car, you can reach many important locations, but not all. Driving in Iceland is relatively easy, apart from a few gravel roads and some parts of Central Iceland. Anyways, you can still get to do these trips, as there are many offers (not that cheap), where they use those huge All-Terrain vehicles.

I've been to many parts of Iceland and have seen much, but not everything...As for me, the best parts of Iceland are Westfjords, Exotic Central Iceland, Snæfellsnes & surroundings, Akureyri, Lake Myvatn, Vatnajökull, Vestmannaeyjar (Westman Icelands - Volcanic Island on the South of Iceland), Grímsey (Island on the North of Iceland), Rejkyavik.

EDIT: If you would like to take your car to Iceland, check Smyril Line.

I've gone to iceland by ferry a few years ago, It was an amazing journey and I'm still planning to go back!
Here is my experience:

We got on the ferry in Denmark.
It would take a week to get there.
You get a two day stop at the Faroe Islands.
There the roads were still made of asphalt (we later discovered that aspahlt isn't that well known in iceland) The landscape was allready amazing.

From there the ferry went trough the shetland islands.
And we got of at Iceland...
We were amazed, we had taken our own car (lancia thesis) and that's as you can see just a normal front-wheel-drive roadcar...
We went all the way round (following road1) and slept at hotels.

I can definitively recommend doing it, just make sure you have atleast a good SUV and preferably a good 4x4 pickup/truck. (rental companies seemed to have just that.)
Not knowing anything about iceland and having found a map at the boat we just went for it.
Getting of the boat it was still looking like faroe and shetland.
The aspahlt soon ended, the whole way round it was gravel and small pebbles.
We encountered numerous landscapes from vulcanic red stone to huge forests and green valleys.
And often didn't encounter anything or anyone for hours and hundreds of kilometers.

We only made one mistake...
After staying in Reykjavik
We went to the Blue Lagoon.
From there we decided to use a shortcut back to road 1.
Road 47 if I'm correct. By now we we're used to the lack of smooth roads and sweet asphalt.
As we allready found this road very rough, there was a redirection.
The road changed to ROCKS the only thing making it a road was that it was a bit more flat and all the big rocks and boulders were pushed to the side.
And after 2 hard slams we stopped and discovered that we had a flat tyre...
We changed to a full spare (luckily) and went to find a small fishing village hoping for a tyre dealer...
We were lucky to find one and our tyre was fixed (as replacing it was impossible...)

Iceland left a great impression and I'm still wanting to go back or even move there.
Just rent a good car...

Our car:

The thing we usually encountered:

I want to point out safetravel.is which is a co-operated government site and should answer all your questions regarding driving safe in Iceland.
I would say that an optimum time limit to observe the island would be two weeks give or take. I would definitely recommend 4x4 for your trip, just so you don't need to take any risks if you feel unsure about the road.

Rental companies tend to be a bit overpriced during the high season so do your research thoroughly before booking. There is also a peer-to-peer car rental site, Carrenters.is, they tend to offer cheaper prices, especially during the high season.

Good luck with your trip!

Driving around Iceland is a VERY good idea. :)

How many days should I spend in Reykjavik if I will stay in Iceland 3 weeks? Is it worth to stay there more than 2 days?

Well... it depends. Yes, there's enough to do and see in Reykjavik to stay there for more than 2 days. However, if you came to Iceland because of the scenic landscape, then you won't see much of that there.

How long does it take to travel around the whole island by car? Not super fast, but if I want to stop at the major attractions.

When we (me + wife) visited Iceland last year we had 14 days to get around the whole island (starting in Seyðisfjörður and ending up in Seyðisfjörður again) and I say it was a pretty good amount of time to get around. But we could've easily spent three weeks as well.

Can I reach the major sights with a normal rental car?

Yes. The "Golden Circle" is very easy to access with a normal car and even more remote sights like Gullfoss are perfectly reachable even with a 2WD. However, the Highlands can only be accessed with a 4WD.

I heard that the main roads are easily accesible with a normal rental car. But if I want to go into the inner regions of the island, I need an all-terrain vehicle. Can I rent such a vehicle? And is it a good idea to just drive by myself if I don't have any previous experience with such kind of vehicles? For example I never used a car to cross a small river.

I think you can drive those roads without previous experience, but I would highly advise against driving all by yourself. If you have to, ALWAYS tell someone where you are going and when you plan on coming back or plan on reaching your goal. I'm sure some saftey measures can be arranged then.