To leave Qatar, whether for a weekend or for good, you need an exit visa and for that you need a letter from your employer saying it's OK for you to go. It's simple in theory, though can be a problem if your employer's in a bad mood or fills in the letter incorrectly. However, it shouldn't cost you anything.
Housing: see the other threads under accomodation etc. At the moment it's very difficult and expensive and changing all the time.