Hello there and welcome to Doha!
You cannot walk your little dog around where the shops are and on the Marina at the Pearl but you can walk him/her along the driveway/road to all the apartment buildings but not on the beach. There is a footpath there and many dog owners in the Pearl are walking their dogs every day. You should pick up anything that the dog does along the way! :)
You cannot walk your dog on the Corniche. It is forbidden. Everyone who has a dog here in Doha (and there are a lot of expats that do!) - we all have the same problem - where to go? All the parks here ban walking dogs too so don't try to do that either.
You can really only walk your dog around your compound villa and it must be on a leash at all times. You can look for a vacant lot to walk nearby your home but there have been some sad incidences where dogs have been poisoned with poison baits - we think they have been put out to kill the rats around as we do live in the desert and there's a lot of rubbish from construction hanging around for them to nest in. So be careful to watch your dog wherever you walk!
You can also drive down to the beach beyond Wakra - some expats go there to let their dogs have some freedom running and enjoying themselves. If you do a search in here (QL) under "Dog Walking" you will see some forum topics.
It is (to me) a sad fact of life living here in Doha that the Municipality of Interior at this stage does not appreciate our dilemma for walking our dogs. And it probably won't change for some time, if ever.
I can recommend two wonderful veterinaries - either Parkview Pet Centre close to Landmark Shopping Centre or Qatar Veterinary Centre opposite the Villagio Shopping Centre. They both have the latest equipment, the best vets and take great care of your dog. Make sure you keep vaccinations up-to-date.
I have a little Maltese, Suzie and we survive walking through the dust, dirt and rocks around our compound. So boring to walk but this is our life here!
Good luck!