Have you asked your current bank? They will/should know more about it than nearly anyone else. Or at least can do research on it to help your specific case.
#1
You could probably manage to continue to use your US account for a while, but that can get tricky quickly. If your account/identity is hacked, it'll take longer to get you a new card and deal with fraudulent charges. I've had problems with this when I simply lived in a town 2 hours drive from the nearest branch.
The bank may also accidently think your real usage is fraud. I've had this happen just travelling across the US.
Also, converting from USD to the Indian rupee might have fees associated with it. A handful of transactions might be free, but continual usage probably will cause problems or fees.
#2
You should probably get whatever standard type of account they use over there. Going into the bank when you get there will be enough for you to get that figured out quickly. Just explain that you are not familiar with their system and they should help you. They may even have pamphlets, like US banks do, to help you figure out which type you need.
#3
A cashier's check might not be the best way to do things. When going through customs, you'll probably have to declare any money you are bringing into the country. Over a certain amount, you'll likely have to pay a percentage and over another amount may simply be illegal or otherwise not allowed. Then again, this may only be for cash and travelers checks, but I would assume any kind of "cash in hand", which is essentially what a cashier's check is, would fall under the same rules.
Also, a physical check could be stolen. A wire transfer is essentially the digital version of a cashier's check, with several more authorization and other security features surrounding it.
You could transfer it through a pair of PayPal (or similar) accounts. Going through a non-government regulated system to move money internationally could raise red flags, though. There's too many people using systems like this to funnel money into terrorist organizations for this to be 100% safe for you.
A wire transfer is probably the safest and most legal way to transfer your money. It might cost a small fee, but your 2nd account should get the money in 1-2 days, be a secure transfer, and go through "proper channels" to keep you out of trouble with various 3 letter US Federal departments.
If you actually have a work visa, or something similar that shows you'll be in the country long term and need significant amounts of money, you should be fine. Keyword: should. If it still throws a red flag and some government official is bored, they may still decide to ask you some questions. By going through a standard money transfer system and having your visa & passport handy, with any other relevant paperwork you have, you'll just need to tell the truth. Unless they are corrupt, they shouldn't have any reason/ability to keep or charge you.
Of course, some of this depends on how much money you are moving. If it's a couple hundred US dollars, you shouldn't have any problems. Even a couple thousand might not cause you problems. Once you get over $10k, there are more regulations and could cause you more issues, but again, going through "proper channels" should show that you aren't doing anything wrong. People transfer tens of thousands or millions of dollars internationally every day and don't have legal issues.
#4
You might have to pay some sort of customs fee for transferring money internationally. This might be charged by the US or India governments, or both. And you may be able to handle it at the bank or you may need to deal directly with the government(s) to take care of it. Again, the banks should be able to help you figure this out, both of them.
Edit:
Just because you have family in India, it doesn't mean you don't need money. I realize the family dynamic there is considerably different than in America, but I'm sure there are things you will need/want to buy that you won't want to ask your family finances to purchase.
That said, here's some info on the different types of account you can get. FYI, I'll be mostly just paraphrasing or copying from the link below.
https://www.axisbank.com/progress-with-us/money-matters/save-invest/nre-or-nro-which-account-best-suits-your-needs
An NRE account is a bank account opened in India in the name of an NRI, to park his foreign earnings; whereas, an NRO account is a bank account opened in India in the name of an NRI, to manage the income earned by him in India. These incomes include rent, dividend, pension, interest, etc.
NRE accounts are exempt from tax. Neither the balance, nor the interest earned on these accounts is taxable. The interest earned on an NRO account is however taxable at 30% according to the Income Tax Act 1961.
Income originating outside India can be deposited into any of these accounts. However, income originating within India can be deposited only into the NRO account. Withdrawals from both the accounts can be made only in INR*.
In case of an NRO account, if the deposit as well as the withdrawal is made in INR*, there is no exchange rate risk involved; whereas, in case of an NRE account, currency fluctuations make you prone to exchange rate risks.
*Indian rupee
What this says to me is that you'd want an NRO, since you can still move money between your US and Indian accounts and it's apparently designed to be used as a regular bank account, however the interest on that account will be taxed at 30%. The NRE won't be able to have income earned in India deposited. Also, with an NRO, you won't be subject to currency exchange rate differences.
