1. Keep the window shut.
2. He kept the shop closed.
(Are both grammatical? Is 'shut' a past participle meaning 'closed'? Does #2 mean "He hasn't run the shop.")
3. Keep the light burning.
4. I'm sorry to have kept you waiting.
5. They kept the man a hero.
6. He kept himself a happy boy.
(Can we use a noun or a noun phrase as an objective complement? Is #5 or #6 grammatical?)
Yes, 1, 2, 3, and 4 are fine. And the answer to both your questions is yes, too!
5 and 6, though, aren't correct. Here is a suggestion to replace 5 --
-- They thought of the man as a hero.
I don't know what you mean by 6, though. Can you explain it?