What to do?

I have seen such questions quite a lot of times, they are everywhere, but I have always wondered if they were correct. Do native speakers find sentences like the following ones natural?

My computer won't turn on. What to do?
There are no screws on the front panel. How to open it?
There are too many trolls here. What to do?

I expect that to be "What can I do?", or "What should I do?", etc.

Yes, they seem natural.

I'm American and they're fine as is, but conversationally I'd use them with should.

They're fine. However, I'd say:

"My computer won't turn on. What should I do?"
"There aren't any screws on the front panel. How do I open it?"
"There are too many trolls here. How do the moderators expect us to keep putting up with it?!" Or something of the sort.

For the most part though, keeping it simple, I agree with Carol.

It's casual, almost a shorthand version of the longer phrase when asking for help. I don't use it myself, but it isn't unnatural and I've seen it in print.
I don't think I've heard it spoken. I agree with Carol and Skippy.

I live in England and I would never say "What to do?", "How to open it?".
I would always say "What do I do?" or "What should I do"