Can I publish an ebook with 0.00$ price?

You can't easily make your book free at Amazon. If you enrol it in KDP Select, you can give it away for a total of 5 days out of every 90, but the assumption is that you're doing that because you want to drive sales on the other 85 days.

What you can do is publish it at another retailer that does allow you to make it free, as well as publishing it on Amazon, and ask your legions of loyal fans to tell Amazon that it's free somewhere else. (There's a link on every product's page on Amazon that's called something like "tell us about a lower price elsewhere".) This is not likely to work unless the book has a lot of downloads at the other site(s), which will cause Amazon to realise they're losing out by charging money for it.

Sure

You just set up your own website and make a download available. Weir's the Martian was originally given away in installments.

None of the usual ebook distributors are going to distribute your work for nothing. Why would they?

If you want to lose every penny.