Multiple questions. No worries. I'll try to answer to the best of my knowledge.
1- Cheapest flying costs? U S A.
If you can't make it to the USA, then South Asia, Philippines and South Africa are some choices you might want to look into.
Type rating helps yes, but does not guarantee you employment. Too many pay to fly guys flooding the international market these days.
2- Employment chances? If you have a green card /
PR in the USA, you will have to start from flying clubs and move your way up to regional carriers (domestic airlines as they are called in India). The pay isn't good but the experience is great. If you get the chance, you might make it to the national carriers. The scenario is quite bleak in the EU. Asia is on the boom when it comes to aviation.
If you DONT have a green card /
PR in the USA or for that matter of any country you are training in, chances of you getting employed are quite slim.
3- Institution reputation in aviation? Your experience and your hours will get you the job eventually. Reputation does matter, but if you are up against candidates with little or no experience, you would definitely be considered. All flight schools train you for a common license (DGCAA/FAA/EASA/ Other ICAO) so as long as you make it through your flight checks, this really would not matter that much.
You may consider flying in India and later on getting a conversion abroad. That way you can save on training cost.
I highly doubt you'd be having a second citizenship since India does not allow dual nationalities. In most countries outside India, you need to be a permanent resident or a national to be able to apply for jobs.
Other forum members will also be giving in their inputs so hang in there. Do your research before you do anything.