I'll try to answer your questions to the best of my abilities. My son is about to turn 8 months old so I had a lot of similar questions when I came out here three months ago.

Toys 'r' Us has a small selection of baby items. That said, I was surprised to see some of the items that I found there (i.e. Bumbo seat) but they were more expensive than what I could pay back home. Brand varies depending on what kind of product you're looking for. Here's what I found:

Baby items (nursing/feeding): Chicco, Avent, Munchkin, Gerber, etc.
car seats/Strollers: MacLaren, Pliko, Chicco, Graco
Jarred food: Gerber, limited selection of Heinz organic, hero
Formula: Similac, Nan (by Nestle)
Diapers: Pampers and Huggies and Luv?? (recommend Pampers)

As far as toys are concerned, you can find them just about anywhere! Plenty to choose from and a lot of award wining toys.

Clothes: For newborns, I'd recommend Baby Shop and Mothercare. Good selection and good quality

Probably the most disappointing thing was the crib bedding itself. I could not find out waterproof sheets so I"ll have to order some online. I was used to really thick bumpers but the ones our here are super thin and some brands only offer bumpers that cover only 1/3 of the crib (just around the baby's head).

Diaper bags tend to be very simple and ugly here. I'm so glad I brought mine.

I can't recall if I saw any baby monitors. We brought one out but since we have the crib in our room, it's not really necessary.

My advice to you is that unless you're really set on gettinng something very specific, buy them in Doha. Some of the baby items are far superior than what I'd expect to find back home (i.e. nasal aspirator). Any questions, feel free to email me at [email protected].