Upvote:2
You'll probably need a visa -- that depends on the particular country. There are no rules that apply in general to "non-Schengen countries" as a group.
Not without particular permission from the country in question. Having a residence permit for Hungary does not even allow you to settle in other Schengen countries for periods exceeding the 90/180 day limit; it certainly doesn't magically give your children wider rights than that.
No. You'll need a visa to enter the UK.