Write Twenty Sentences on the Board

[Soft sign]

I believe Russian does fall into the category of languages that uses the informal (ты) when an adult speaks to a child, regardless of relation.

Yes.

I......... would think you would say "write onto something" with на + acc, otherwise you're standing on top of the board while you write. Could be wrong.

Написать на доске (prep.) is correct.
One can say переписать из книги на доску or maybe выписать на доску, but not писать/написать. Dunno why.