How long can a person stay at her/his house without paying the mortgage?

who will get the right to foreclose?
first mortage?
home equity?

should she/he send the house key to first mortage or home equity?

I would stay until the foreclosure auction has been scheduled. It will be very unpleasant but they can't make you move until then.

I would also suggest that you would be much better off if you try and make the payments, try and sell the house if you can't make the payments, and contact the first mortgage company about a compromise sale (short sale) if I can't sell it any other way. Failing all of that I would contact both mortgage companies about handing them the house back "in lui of foreclosure". This is just slightly better than foreclosure- not much. Only advantage is that there are fewer costs that will be applied to the lien they might file against you later.

The Wall Street Journal just reported about one homeowner who lasted 11 years before the house was taken, and is still fighting it.

The mortgage holder has the right to foreclose. That's one of the problems popping up right now. Many mortgages have been packaged into composite securities and sold to investors, but title to the mortgages was not transferred in the bundling. A court ruled that the mortgage holders cannot foreclose because they are not the owners of record and the owners of record have no reason to foreclose because they sold the mortgages and do not have a legal interest. So a lot of mortgages are suddenly in limbo.

Normally either the first or second mortgage holder can start foreclosure proceedings. Presumably a home equity loan is subordinate to the first mortgage. However, an owner may default on one loan and continue paying the other, so the owner of the defaulted loan will act.

Until the buyer at auction shows up with a deputy sheriff & whatever notice or writ your court issues telling you to git.