How much would a broadband cost per month?

200 QR per month.
I prefer eating veg food. Is that a problem?

No problem at all

How is the night life there? Any decent pubs/clubs to hang out?

No night life in Doha,

How about renting a car for monthly use?

Yes u can do after obtaining a driving licences, it may range from 1500 and above per month.

How much can I spend on petrol and travelling if my office is not too far off?

Petrol is cheaper than water, so not much.

How is it for Indians out there? Any cultural/religion based
problems?

You will find plenty of Indians, no issues.

How are the recreational facilities? Like movies etc?

You have decent theatres here, you will get to see new hindi movies as well.

How much can I expect to spend on food/groceries? I do not know how to cook(yet).

Food on an average 50 QR per day.

What are the tax rates in Doha?

No taxes of any kind here.