I (EU also) had to pay for a visa. But it isn't a lot of money, and they take a credit card. The main difficulty I had was getting the immigration official to understand my answers to her questions - her English was only slightly less retarded than my Arabic. In particular, we kept getting stuck on the "How long will you spend in Qatar?" question. When I answered "a small number of hours", that generated the <point to transit area> response. Getting her to understand that I had only a few hours between flights, but still wanted to enter the country, was challenging.

As they have free wireless, you could cruise the internet. But my experience was, the free wireless wasn't all that reliable.

The Qatar website has some option to purchase brief tours designed for stopovers. They're probably a rip off, but they would be easy, I guess. Of course, the timing of your stop has to line up with the timing of the tour.

There's a mosque across the street, with a relatively rare patch of green in front of it. Just walk through the parking lot. Unless you're really into mosques, though, that would probably get old before the end of your transit.