This is the kind of swag I like!

Depends

You should expose the conditions behind that wall.

Remove and store the stove. Take the whole thing apart one layer at a time. I think your primary concern should be the electrical. Try to carefully expose any wiring. You might want to hire an electrician to trace the circuits and de-energize the space.

From your description, the odds are in favor of removing the entire structure. You might be able to salvage the building materials.

I’d be most concerned about the electrical work. Consider adding a combo smoke/CO detector ASAP.