Hi SumiVidya
Congratulations and wish you a very happy and blessed married life.
And welcome to Singapore.
I will try to answer your Questions one by one.
1. It is NOT true. Except workpermit holders(inclusive of demestic helpers) all the dependant passholders and permanent residents are allowed to deliver here.
2. You will continue to work in your DP because dependant pass holders are allowed to work in Singapore unlike US. Your pass status will not change unless your husband's type of pass changes. You dont need to return to India.(But take note that delivering in Singapore or India doesnt make a difference as the kids born in Singapore will not be automatically getting Singapore citizenship. They follow the parents' citizenship only, as dependants of the main pass holder.)
3. I do not know how far it is true because I have relatives and friends here- bachelors who got their
PR before marriage.
I hope that my answers have eased your worries a bit. Please go through the ICA website of Singapore and the forum for expats in Singapore to get full idea of the rules about the visas and passes.
All the best.