What is the maximum cash gift a person can receive from Indian
There's no limit on what a US citizen can receive. The only thing to keep in mind is that if you get $100K or more in gifts/inheritance from foreign persons - you'll need to report it (not pay any taxes!) to the IRS on form 3520 as part of your tax return.
Beyond what limit the cash received from parents are taxable?
Why would it be taxable? If it is a gift - it is not taxable.
Can stocks/shares held in the name of Indian parents be transferred to
children living in U.S and what is the tax implication on it ?
It doesn't matter what you get as the gift, the $100K rule I mentioned above applies (if you don't receive the actual money - then you need to calculate the fair market value of the gift to know if it needs to be reported).
Other incomes like rent/dividends received in India, should the tax be filed in India or U.S.?
Most likely both. US taxes worldwide income, and India is likely to tax income sourced to India. There's a tax treaty between India and the US that may regulate some cases, but for most parts it doesn't apply to US citizens so you need to check carefully. In any case, you use the taxes paid in India as credit towards your US taxes on the same income, and that allows to alleviate the double taxation to some extent.