As long as it doesn’t have any plastic parts, it should be fine. Many pans are meant to be put in the oven. I bake cornbread in cast iron skillets all the time.

Depends on the materials and temperature.

Teflon and plastic? Over 350F does bad things. Teflon and stainless steel? Probably safe up to 500F.

Cast iron or all stainless steel. Not a problem.

Check the manufacturer’s website for better guidance.

Oven-proof frying pans are a thing! I have one that I use for my giga-frittata all the time.