It seems to me that if a house exists, someone owns it, unless you consider government possession NOT ownership.

Even if you argue for the ownership of a house, the land it sits on is ultimately owned by the state, so I don't think that's a very productive topic...

So if the government possesses the house, they should provide it as housing for free to someone, right?

Not necessarily for free (although, as I stated, that would be ideal), but certainly not for profit.

And a person CAN buy the house, but if that person is not going to live in it, he should provide it to a person to live in either rent free OR at a price that is not more than the taxes and costs so that it is essentially provided non-profit. Correct?

That would be incentivised, yes.