Any of these are valid constructions, but they mean different things. To swing would imply that the child is moving back and forth. To hang implies that they are just holding still. To dangle often implies that the thing is not very securely attached, or in a dangerous or temporary situation, so it's probably the least likely word one would use here.
"From" is the correct preposition.
As a native speaker, and father of two small children, I would say that the most likely words I would use are "my child likes it when I let him hang from my hands", to emphasize that I'm doing the work. Or "my child likes it when I swing him while I hold his hands".
But those are more complex sentences. "My child likes to hang from my hands" is pretty good because it leaves it ambiguous who is doing the work, and the listener will just assume it's you. Or "My child likes to hang from my hands and swing" does the same thing for a swinging action.