I had managed to make a old version of Firefox to treat URLs entered in the location bar as relative. In my opinion, this is the reasonable way to work. The computer doesn't have to guess what you mean, and does not have to guess if the TLD is valid. It will do what you typed into the computer; and if it is an error, will display the error message.

For searching, I had done that if you put a colon at first and then the name of the search engine, then it will search.