Well as far as delivering in Doha; if your parents are here and have a RP here, its not a problem. My parents too are in Doha and I came in the month of June on visit visa when I was 5 months pregnant and delivered here at Hamad. I don't think it should be a problem. You just need to produce a copy of your attested marriage certificate when you register at Hamad along with your sponsor (in your case, your dad's) details. I also got my baby's passport made here at the Indian embassy. As far as renewing your son's passport, check if you can get it done in UAE before you come here as you say you have a month. Here in Doha, the Indian Embassy is pretty prompt and it takes 3-5 working days only. UAE should be the same I guess.
Anyhow I hope I have helped in someway. All the best!