Airport to downtown is fairly quick, you can check google maps but my recollection is 15-20m. Most hotels and some restaurants are able to serve beer, so that should not be a problem. Doha is not really a walking town, so I would really choose your hotel based on price, comfort, food, perhaps even facilities (the Four Seasons has a great pool area) rather than location.

excellent, thank you for this info, I have a free night cert for Hyatt and was thinking of burning at the Hyatt Doha when I noticed that it was one of the furthest hotels from the airport. That said, I too have an 8am flight so I'm assuming traffic at 5:30-6am isn't going to be busy.

Interesting reading this thread, certainly from 2013. We moved here in 2015 and the place is growing rapidly. QA have started issuing free visas for stopovers and here's a few pointers for anyone who is thinking about stopping over here:

Airport to West Bay - 15mins
Airport to The Pearl - 25 - 40mins traffic dependent.
You CAN drink, no issues, bring your passport though. Any of the good hotels have bars; Sheraton, Marquis Marriott, Shangri La, Rotana (City Centre), Four Seasons, Hilton (West Bay & Doubletree) all have good bars in them.
Quick hit list of where to visit:

Close to airport:
Soukh Waqif (wander through old back streets, fabrics, spices, usual souk fare)
Museum of Islamic art
The Corniche - essentially a 6km semi circle beautiful for walking along, nice parks along the way etc.

Mall of Qatar has just opened but this is an hour away, but is by far the best mall here.

The pearl Qatar (low key version of The Palm in Dubai) lovely food, drinks (no alcohol apart from the Kempinski hotel)

Well worth a stopover now in my opinion. Sorry to hijack the thread, any specific questions DM me, more than happy to help.