If POA covers finance what covers vehicles and apartments that are owned etc.?

You should look over any Association Policies; my parents live in a Senior apartment building and even though they "own" their apartment, they cannot rent it out to any unrelated persons, and they also cannot let people under age 55 live their if they are not also living there.
I suspect from what you describe, the co-op has some type of stipulation about a minimum age, or no sub-leasing (sometimes you cannot sub-lease even to relatives, there is no provision for anyone but the owners living there).

Your mom should be looking at the ownership papers on the apartment to find out who can live there if the owner is no longer there. Can she sublease it to you? Does it have to be sold? Is there an age restriction on who can live there? Good chance you need an elder care attorney to deal with the property issue and whatever else is likely to come up if she dies intestate (without a will).

The person with the Power Of Attorney can sign for property and other things unless the POA is limited in some way. Most financial POAs assign the right to transact all business to the person named. You mom is the one who should handle this.
Good luck,
Carol