1. Wesley was allowed to apply at 14. I don't think anyone mentioned that as being a minimum age, though. In fact, I would suspect they would probably just trust that the entrance exam is clearly too difficult for extremely young people, since any sort of hard age limit would be problematic to enforce across different species (Imagine telling an Ocampa they can't join until 14...).
2. Having kids on board seems perfectly normal in the 24th century, though with as little info as we have, it may be the case that extraordinarily small ships or ships on certain types of missions don't allow families. And, just for the record, that's specifically the 24th century. In the 23rd century, they didn't do that sort of thing at all as far as I can recall.