You can if you're referring to the notion of travel in general, rather than a specific instance of travel. Used as a noun, the word "travel' is uncountable, so it doesn't appear with articles. E.g. Travel is cheaper than it was fifty years ago.

Yes, you can. You can say: I’ve experienced many things during my travels.

You can say "my travels", if you are talking about your past trips and adventures. (It has to be plural in this situation)