Property, in the abstract, is what belongs to or with something, whether as an attribute or as a component of said thing. In the context of this article, it is one or more components (rather than attributes), whether physical or incorporeal, of a person's estate; or so belonging to, as in being owned by, a person or jointly a group of people or a legal entity like a corporation or even a society. Depending on the nature of the property, an owner of property has the right to consume, alter, share, redefine, rent, mortgage, pawn, sell, exchange, transfer, give away or destroy it, or to exclude others from doing these things, as well as to perhaps abandon it; whereas regardless of the nature of the property, the owner thereof has the right to properly use it (as a durable, mean or factor, or whatever), or at the very least exclusively keep it.
Yes it is permissible as far as the ownership is properly obtained. The rental income from such properly owned or even properly mortgaged will be then permissible as well.
Yes it is a Waqf if it was said orally by the person who was the owner. Registration is good but not a condition for validity.
You must keep on trying to reach to the real owner and return the parcel to them. If -after trying your best- you become sure that there is no hope at all to reach to the owners, then the item will be Majhool al-Maalik (unknown owner) which should be given to the poor on behalf of the owners.
Yes, it is permissible to purchase a share in a property with any other partner or partners.
Salamun alaykum wa rahmatullah
The brother and sisters would not inherit because they belong to the second category and as long as any member of the first category exists, members of the second category would not inherit.
In the case of your question, according to the fatwa of Ayatollah Khamenei, the wife would inherit one eighth of the assets and the rest would be divided equally between the daughters.
The value must be based on the current value and not the cost price. If any one from the heirs wants to buy the share of others, he must pay them according to the current value of the property.