Possible Smoke Damage?

If it's getting smoke on it then that's not good. Smoke creates dust.

i yes teh snoke could be damaging your console. Any kind of heat is bad for the xbox.

As stated ensure it is well ventilated. It depends on how far from the kitchen the console is. Grease from cooking is more likely to cause an issue as it will build up inside the vents, fan etc.

That I would say if your kitchen and the room in question is both well ventilated I doubt there would be any damage, unless we are talking literally out of control fires. But yes try to keep your console in a cool well vented area. If you're cooking it
might be better to leave the console off.

It's possible, but my question would be how much smoke are you making? What are you cooking? If you're generating smoke, it's going to rise to the ceiling, and though it could add particles in to the air, the amount of particles that would need to accumulate
to cause an issue in an adjacent room (unless your console is stored on or near the ceiling, without any ventilation between the cooking area and the console) would have to be a large amount of smoke or a accumulated deposit over time, which could take years,
but again depends on the volume.

Are we talking about making delicious pot stickers and the oil generating some smoke or are we talking about you messing up a hot pocket in the microwave? That doesn't particularly have anything to do with your consoles health, but I'm just curious what
you keep burning.