I assume it is CIBIL Score and not CRISIL Score as mentioned in the Heading.
1. Private Banks have very tough Loan Approval Process. With PSU banks you can negotiate. Carry all your IT papers showing income proof. Also higher the down payment you are making for Home, higher the chances of approval. NEGOTIATE is the Keyword since you are a defaulter according to CIBIL report.
2.Yes they can close the credit card account if they want to, it is their internal decision and no RBI rule can restrict them from doing so.
3. So bank did inform you about closing your credit card account. Did that letter inform you about being reported to CIBIL ? Start writing to the bank saying that according to RBI's Guidelines for CC operations under clause 6.2.c they were supposed to inform you, but they haven't.
Before reporting default status of a credit card holder to the Credit Information Bureau of India Ltd. (CIBIL) or any other credit information Company authorized by RBI, banks/NBFCs should ensure that they adhere to a procedure, duly approved by their Board, including issuing of sufficient notice to such card holder about the intention to report him/ her as defaulter to the Credit Information Company. The procedure should also cover the notice period for such reporting as also the period within which such report will be withdrawn in the event the customer settles his dues after having been reported as defaulter. Banks /NBFCs should be particularly careful in the case of cards where there are pending disputes.
The thing is they will get to know that you are desperately looking for Home Loan and want to settle and get rid of the previous bad remark. They may not move an inch but NEGOTIATE, NEGOTIATE and inform them that you want to settle the remaining at some modest amount and they should update the status as "A/C Closed" leaving the status field blank, that erases of "Written Off".
Go through other posts on the Forum about CC Settlement and you will know.
Best