“What age should a child use disc brakes. Yes, the stopping power is great than caliper brakes/rim brakes. But are there fears for heat.” “If a child rides a mtn bike, will he burn themself on disc?”
I think the better question to ask yourself would be, does my child know how to ride and operate a bicycle safely?
Lots of moving parts on a bike to potentially burn, sever, pinch or trap skin, digits, loose clothing, long hair etc., if you think about it?
Why not teach them or get someone who teaches beginner bike use and safety, to teach them the basics on safe operation and obviously use of a bike, off road in a school yard or playground first, progressing to on road use.
Stuff like checking your bike before use, then coordination and balance using cones and numerous games and exercises to learn using gears appropriately, moving of safely from a stationary start, signals, turns etc.), into leading to on the road practice of most things first practiced in the playground, but now including moving off safely into the flow of actual traffic, positioning for visibility for you and other road users benefit, real signals and turns, junction protocol etc.)
When I was growing up the gov. dept. for safety on the roads had us doing this in primary school.
To answer your question specifically though, yes discs can get hot and burn you if you are not careful, they can also slice fingers/hands if rotating or just by not being careful when tightening or loosening wheel axle quick releases.
Stopping power is variable on some of the children’s bikes I’ve seen with discs, depends on the set up and also quality of the components, wear and tear (abuse) they’ve received from the owner but also the environment (stored inside or out?)
“or should they go with a rim brake without braking heat?”
Again, it’s not about the brake, it’s about whether it actually works appropriately for it’s intended use. Children’s bikes can be pretty clunky though, with cheap components that often do not work as well as the adult versions, so again, just make sure when buying and testing rim or disc equipped bikes that they are actually set up and work properly. (i.e. stop you when you want to stop!)
Proper use of front and rear brakes will make a big difference so again it’s back to learning to ride safely and in control, including when the need for emergency braking arises, (balance, coordination, practice etc.)
“Should a 24 inch or even 20 inch wheel bike have cable pull disc brakes or rim style brakes.”
The few different brands I’ve seen mostly had rim (‘v’) brakes, and when adjusted worked well for the children using them, (size and weight of the child and use/abuse of the intended bike. But I have seen a couple of brands with disc but they were the exception.
From your questions maybe you didn’t ride bikes a lot as a child, but it’s never to late to start now and have some fun alongside your children, so why not include in that, learning and instilling in both you and them, the safety and technique aspects mentioned above for safe, enjoyable, more efficient riding.
Don’t overlook gaining some basic bike maintenance self-sufficiency, to allow you to keep the bikes rolling along at all times, it’s not that difficult to achieve and easier than you’d imagine.
Have fun.