This could depend entirely upon the specific words used in the sculptor's will, which framed the gift to the museum. However, absent specific bequest or a personal property residuary clause to the museum, some other heir of the estate was the lawful owner. That person may have "abandoned" the property by failing to remove it from the premises prior to conveyance.
It would be a fine thing to offer to bequeath it to the museum collection, in exchange for some simple recognition (not to mention a potential tax deduction for the charitable gift).
It is owned by whomever owns the land it is on. If the land is considered a public road, the county or community will 'own' the property. Others may have right of way rights. If the deed doesn't specify a right of way, there may have to be some legal action to designate it as such, or compensation paid to the land owner. Consult an attorney and you may have to pay for a survey.
If they "win" the house then they can keep it until they fail to pay the property taxes and can't find anyone to help them keep the house.
Real estate is located officially in one state or another, so the transaction would be covered by the laws that apply in that one state or the other, or preferably both, if you pay your lawyer to do it right.
No. Dong so may cause adverse consequence if and when one of the two parties move. It is your property and therefore your responsibility to fence your own yard. You can share the fence and the costs. You would each pay half. Also, depending on the laws where the property is located. If your neighbor puts a fence on your property and not on the property line (a few feet inside on your property) they could end up owning those few feet after several years. Adverse possession.