I will answer this in a few parts
I assume you are looking for some kind of a web release for what ever media you are recording and I assume you want to go high def. For this most if not all of the current DSLR's from the big makers (cannon, nikon, etc) will shoot in both 720 and 1080 (i and p depend on the camera but for what you are looking to do I would say either will suffice). When it comes to video my only requirement is that the camera has an external mic input (I know the Nikon D series has them). The on board mics on DSLR's are not that great and an external one is a must. This would be your main camera.
Go Pros are GREAT and come with a wide variety of mounting options. They are also in waterproof cases (I have used mine in the pool a few times with good success) so I would think they could stand up to some puffs of steam although continuous steam will be an issue for any camera.
You can also troll ebay or tag sales etc. for used DSLR's for your overhead shots. You can usually find them cheap and be less worried about them breaking form steam/heat
Good tripods area easy to find. In this case I would look for something hefty but on the smaller side. I have a small Slik 504 that is small but really steady and has a smooth head. I also have a MeFoto for when Im on the road. These are much smaller than the Slik's but REALLY well made (all aluminum) and will not have a problem holding your DSLR.
Theater/Video lights can be pricy as can their mounting hardware. For this I will offer a simple home brew hack. Go to home depot and pick up the double halogen with stand work light. They are constantly on sale and fit the bill really well. they are bright, cheap, and come mounted to a stand. They usually are capable of pivoting in all directions as well.
I dont know much about this issue but one possibility is to mount a mirror over the stove and shoot the mirror. This puts the camera out of the line of steam and out of harms way. You can also get one of those shower mirrors that does not fog up in high steam environments.
Hope This Helps...