If I recall correctly, a parent is likely to use the less polite ~なさい (polite command) form.

It marked this kanji sentence wrong because I typed 野菜（やさい） in kanji. I've reported it (Jan. 5).

I think in this context it rather means Please eat vegetables' in general. If some specific vegetables it may say この・その　野菜、because 野菜 is too a vague expression and if you refer some vegetables on your plate, you may say にんじんを食べてください Please eat the carrot. etc...

I think Duolingo requires literal translation word for word, and the word for "some" is not here.

The て-form of verbs lets you express sequences of actions, but it also has many special idiomatic uses. In particular, with ください it's a way of making a request to do that thing. (This is common enough that if you leave out the ください, it's basically implied.)

Sure.