Yes, you may purchase a home with cash while you are receiving social security benefits. Your social security benefits will not be affected by purchasing a home.
You should keep records for your own financial records, and for future tax implications, if the house ends up being sold. Purchasing a home does not trigger immediate tax responsibilities, but you do need to keep in mind that there will be property taxes that need to be paid each year that are assessed based on the value of the property.
Regarding the rental income, yes, you must report all rental income and expenses on your tax return. Rental income is typically taxed as ordinary income and reported on schedule E of your personal 1040. You can deduct rental expenses such as mortgage interest, property taxes, maintenance, etc.
Here is an article regarding how to report rental income & expenses in TurboTax:
TurboTax Where do I enter income and expenses from a rental property?
