the answer for part 1
1 Sale Proceeds 75,00,000
2 Cost 7,00,000
3 Index cost [750000*1125/591] 13,32,487
4 Long Term Capital Gain [1-3] 61,67,513
Tax on above @ 20% 12,33,503
If she purchase or construction a new Resi. House then
5 Amount Paid 30,00,000
6 Net Long Term Capital Gain [4-5] 31,67,513
Tax on above @ 20% 6,33,503
Part II - She need to keep money in "CAPITAL GAINS ACCOUNT SCHEME" in any nationalised Bank.
Part III - the better way that the new flat will be purchase in the name of your mother and she will pay the full amount. the she will claim the deduction of Section 54.
Part IV - As she earn the capital gain and claiming the deduction u/s 54 so she required to file the ITR it's don't matter that she is home maker and she does not have any regular income.
and you inform that she will deposit the 50 lac in FD. So she will earn the interest on FD,
Every year Bank will deduct TDS and your mother is required to file the ITR.