My father just recently passed away and a child has presented themselves as his son and is requesting property rights. Can this be done? There has been no established paternity.

It can be done, but it is incumbent on the petitioner to prove blood relation to share in the estate. I assume your father died without a will. It will be up to the probate court to determine if his petition has merit.

Depends on the law in whatever state in which you are located.