The first pay payment after closing is normally 35-40 days before your first monthly mortgage payment is due.
You may buy additional months by prepaying the interest. Make sure your loan officer know this fact as all financial information would be enclosed in the mortgage loan doc you would be required to sign at closing. Failure to have the additional prepaid interest and mortgage start date would be cause for new loan docs and additional escrow funds for the closing to take place.
You might consider beginning the process around the middle of April for your pre-approval with a mortgage lender. This pre-approval would give the amount you are able to pay for a house based on a formula used by the mortgage underwriter to determine this amount.
Your pre-approval would be good for approximately 90 days. Failure to find a property prior to the expiration date, you would need to be issued another pre-approval, this issuance in normally not a problem.
A lease/rent with an option is normally done by individuals not banks. These type agreements almost always are in favor of the owner not the person leasing or renting. Beware!!!
I hope this has been of some benefit to you, good luck.