Indra
I am asking the same questions being due in Oct, but I have an EP myself. The situation, however, is the same for both Ep and DP.
Before you go to the hospital for the delivery you will need a doc from your embassy stating that they accept citizenship of the baby. It's a standard form and you shouldhave no trouble getting it. All the other info above is correct. Without it certain nationalities can have trouble being allowed to give birth, I am told, though I find it hard to imagine the authorities putting a 7cm dilated woman on a plane home!!
Passport can only be issued after the birth because, as you say, they will need a photo and the name of the baby!! Also DOB! I have been told there is a photographer who comes round after the baby is born and takes photos for you with a special camera mounted above the baby's cot. I am planning on getting several sets of these photos just to send to rellies overseas as well as for all the paperwork etc.
When my friends did this, they got the birth cert first, then passport, then DP. Though it would be very useful to get birth cert and DP at the same time! You also need to give your own embassy the Sgp birth cert so they can issue you with your own country birth cert, done at the same time as the passport.