Seems like we're looking for the same thing around here. I'm pretty new here in Qatar and I've been in search for a lap pool nearby where I can continue my training...or at least keep myself fit. After reading several posts here, seems like those that are easily accessible (for a fee) are hotel pools.
As for running, the only route I've tried so far is at the Corniche. I usually start near the Sheraton hotel and go the opposite end then back.
Biking on the other hand...haven't tried yet as I didn't bring my bike here. My friends advised me not to since it's not a safe place to bike - due to the hot weather and also crazy drivers on the road. Although I'm still planning to buy a bike eventually and look for a long stretch of road outside the city where one can go cover several kms during the cold season.