I would suggest contacting an elder law or estate planning attorney in your area. There may be a risk in doing this, and there may be better ways to accomplish your goals. The Find a Lawyer tab at the top of the page allows you to search by practice area and location.
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You may certainly pool your money and buy a home with you listed only on the mortgage - assuming that you qualify for a mortgage on your own.
As to the second issue, i.e., protecting assets in case your mother will need long term care, as my colleagues have suggested, you should consult a local estate planning / elder law attorney who can best advise you and your mother as how to best protect your assets.
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