For a dog, I would usually say "trained." I train my dog to stay out of my tulips. I might, possibly, say "that dog trampled my tulips again, I'm going to have to educate him" -- maybe turn the hose on him? For a human being, either one. You can train your children to behave properly, or educate them in proper manners.
No. Hit'amen is used for physical training or for practice (not necessarily a sport - you could be practising a musical instrument or chess).
The verb חנכ, in general, and this particular form, in particular, are used in general for education but are more properly used for learning proper behaviour rather than learning new information.