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1. Would there be a gift/estate tax on such transfer?
No, not for you. You would need to report the gift if it is over, in value, more than $100,000 using form 3520.
2. Once transferred to me, what would be the US rate of tax on long-term capital gains arising out of sale of such stock in India? Would inflation index be used? Would the period held by my Dad be added to the total period?
That would depend on what you holding period and adjusted gross income in US was at the time of sale. Only your holding period would count and if congress does act to keep the current rates you would have Capital Gain form 0% to 15%.
3. Considering that most of the stock is long-term and there is currently no tax in India on long-term capital gains (unlike US), would it be beneficial for my Dad to sell and re-purchase the stocks before the transfer to me?
You will have the same basis in the stock as your father has so that would be the only thing that would change. If it made his basis higher then that would benefit you.
4. If my wife converts her visa status from H4 to F1 (student) and then the shares are transferred to her instead of me, would the sale of such stock by her be exempt of US tax on long-term capital gains? If she then transfers the re-purchased stock back to me, would that be subject to gift-tax? Any limits that apply?
Nonresident aliens (valid F1 VISA holders that have not reached their 5 year limit on counting days in the US) do not pay US tax on foreign assets sold. You would need to report the gift from a Nonresident alien just as with your father's gift.
5. If my wife holds F1 status for only a portion of the year, would she still be considered a non-resident alien during filing?
This question is outside the context of the original subject but, she would be a Dual Status and need to file 2 US returns reporting the income under residency and nonresidence periods using 1040 and 1040NR.