My fiance's parent purchased a home for us in full. The deed to the home is in her name only. I pay all RE taxes. Can I deduct this on my single filer income tax return?

Only the person who is legally obligated to pay the property taxes can claim them. If your name is not on the deed/property tax bill, then you can not deduct them.

Thanks Carl. That brings me to question 2... I owe the IRS back taxes which are filed as currently not collectable. If I were to be added to the deed/property tax bill, could/would the IRS go after the home's equity in some way? The equity is for our future children so that's the last thing I want.

Absolutely. They'll put a lien on the property at a minimum, and then you're stuck. If you sell it at a gain, the IRS gets their cut first. If it's willed to beneficiaries, the IRS gets their cut before any distributions are made.
You really "need" to clear up your tax debt ASAP, as I can assure you that even after death, it will NOT just go away.

Understood. I figured as much. Thanks Carl!