Hi Arun,
Nice doubts buddy! Given below are the solutions
First , on re investment of capital gains in new House property, to claim exemption u/s 54, purchase of HP should be along with possession of the HP, registration of document is not necessary, POSSESSION IS IMPORTANT.So if you can prove to the AO that you have the possession of the NEW HP purchased, you can claim the exemption.
Next , if the new house property is sold within 3 years the amount exempted earlier will be reduced from cost of acquisition of that house property and the assessee will be liable to pay STCG and NOT LTCG (within 3 yrs is less than 3 yrs hence the HP is short term capital asset)
Finally, you ask that if the sale proceed on sale of HP within 3 yrs is again re invested in New residential house then is sec 54 exemption available ? answer is YES SEC 54 EXEMPTION IS AVAILABLE. And therefore if he avails exemption he will not have to pay any tax on the sale proceed,
Hope this helps to clarify your doubts!
GOD BLESS!