The following are my opinion based upon research/analysis. Since this involves Taxation rules, suggest you consult a Tax attorney for their views.
a. Yes. You can transfer any any amount of dollars to your NRE account without paying any additional taxes.
b. Since you have passed Substantial Presence test, you will be categorized Resident Alien. You need to disclose the interest earned on your NRE account. This amount will be clubbed with your total income in US. You would have to add this interest income in your total income and calculate tax thereon.
C. Transferring money to Parents? Answer is Yes /No. You need to elaborate on this further. Advantage of parking money in NRE account is that you can repatriate money in dollars any time you want. Assuming you need to buy a house or further education , this would be useful. Incase you don't have any of those goals, Yes you can transfer money to your Parent if they are in need of funds.
d. Any transfer above $14000 will attract gift taxes. This $14000 is the overall limit for a Tax payee. Incase you transfer $14000 for each parent, then it comes to $28000. There are chances that IRS may raise a Redflag that you are circumventing the limit by splitting the recipients.
e. Only when you transfer more than $14000, you need to report and pay the gift tax.