Definitely put it outside.

VFTs require full sunlight. They're natural habitat is in North America, so your climate should be perfectly suitable for it.

Since it's outside, you probally won't need to feed it, especially not hamburgers . . . so don't worry about that hah.
I'm sure it will catch some nice bugs on its own; it's there job ;).

And no, it doesn't "sleep" at night. I'm not quite sure why it would droop at night, it could because it needed more sunlight, I wouldn't be surprised if you found that venus fly trap inside, which is a big no no.

On that note, if you did find it inside, and in one of those plastic domes, you will probably want to aclimate it to lower humidity and more sunlight.

Hope it helps,
Samuel