'Allow for' in this case means 'planned for' or 'set aside' the two days. In the event that you mentioned, two days wasn't enough!

Yes, in this context, "allowed for" means "planned for" or "allocated time for." The sentence suggests they scheduled or set aside only two days to spend at the amusement park, but it turned out to be insufficient because the kids enjoyed it so much.

"For two days" is a adverbial phrase. It is how "we allowed". You could equally well say "We allowed two days." If you did it that way, "two days" would have been a direct object, a noun, and the thing that "we allowed". So what's the difference in meaning between the two choices? "Two days" is more specific than "for two days". "For two days", being less precise, can be interpreted in more ways. For example, it could mean that you created more free time, or saved more money, or arranged your schedule. "Two days", on the other hand, merely specifies the number 2.

Yes. When planning the itinerary for the holidays only two days were set aside to visit the amusement park. The other days were for sightseeing, visiting museums, lying on the beach etc.