There are no restrictions on the clothes you wear, just that women are advised to dress modestly.
During the month of Ramadan, some restrictions exist - restaurants and bars are shut, etc., but that is also the time when most expats leave Qatar on vacation.
It is way too hot then, so it's best you go home and enjoy Ferragosto on a nice beach :)
(I'm quite new to Qatar, and haven't yet seen summer or Ramadan here - this is only what I've heard.)

If you're a sports fan, you'll find many more things to do. There are clubs/associations and informal groups for most sports here. You could make friends off those, too.
Remember, social life isn't very easy here, so use every opportunity you get.
The key to enjoying your life in Doha is having a nice set of friends. However, it usually takes time. Find a place to start - your workplace, a sports club or even QL.
I've made some friends off QL, as have a lot of other people.
There are some nice restaurants, cafes and bars, but don't expect a roaring nightlife.

Oh, and driving is something you'll have to be careful about :)

The rest you'll figure out as you go along. Life is a little slow, and most things take a little time. Devi prendere le cose con filosofia ;) That apart, it's not a bad place.
I'm sure you'll be okay. Buona fortuna! :)