You may want to apply for a mortgage now so that you find out how much you are pre-approved for and what your monthly payments would amount to. Since you will be quoted the max you can borrow, this helps you decide how much house you want (realistically) to buy to keep your monthly payments comfortable. This will also allow for you to produce without haste, the documentation your lender will require.
Finding the right house depends on each individual. In your scenario, you would have 3 months which is excellent. Once you find the right one and go under contract, it may take a minimum of 30 days -typically about 45- to close.
In your scenario, closing in July seems ideal. Your first mortgage payment would start in September.
Lease back is for sellers, not buyers. Perhaps you're referring to lease to buy which is typically done with a higher-than-market rate, to people that are not qualified for a normal loan. I've never heard of banks offering delayed payments deals.
RE pro in Texas