Self drive certainly not recommended in India - lots of cars with drivers you can rent for the whole day including 4 wheel drives.
Better to fly between Mumbai and Goa if you only have 2 weeks vacation.
September is the ending of Monsoon and some days can be very wet - not the right month if you want endless sunshine - leave it until October.
Hotel accommodation not expensive - you may find the size of villa you would require will be about the same price as a hotel - home rental is not common in India and would only be available in Goa not in Mumbai.

These sites may be useful, to give an idea of the type of villa acommodation available in Goa

http://www.holidaylettings.co.uk

http://www.lazydays.co.uk

Can highly recommend Lazydays