USA expat needs 1BHK or 2BHK fully furnished flat to rent; Long Term

I have very specific constraints an an apartment I need in Mumbai (and areas). I may not have been as clear as I should have been on recent posts. I hope the below listed constraints are clear and concise. My time is ticking with 73 days left as of this posting. I am arriving on 23 January 2015. I must be in an apartment/flat not later than 1 February.

I have been told that being on a tourism visa that I can not get a "registered" apartment. This is incredulous.

1. I am coming on a ten year tourist visa
2. I am not working and I will "not" be working.
3. I have a hard budget of 30,000-40,000, maximum.
4. I cannot and will not pay more than two months of deposit.
5. Based on a variety of rental agencies I've seen on web sites (unless they're highly inaccurate) with fantastically furnished apartments all for under 38,000, I would like to look in the following areas:

Andheri East
Andheri West
Bandra East
Bandra West
Chakala
Chembur
Powai
Juhu
Khar
Kurla East
Santacruz East
Santacruz West

  6. Is any or all of the above doable?
  7. I do not want and can not afford any surprises after I get to Mumbai.
  8. I can provide superlative references from landlords in New York City.
  9. I am not married, I am a single male.
10. Can anyone help? A firm Yes or No will be appreciated and necessary
11. Very serious replies only who can reply positively to EACH point above.