I would like to have a shortcut to find the next occurence of the selected word.
Maybe this could be realized when you have a word selected and press the "Find Next" key, that it will look for the next occurence of the selected word.

And also the same request for "find prev" when a word is selected.

Try the F3 key.

It would be nice if F3 does this.
But when I search (CTRL+F) for "foo", and later select a piece of text like "bar" and press F3, my request is that it will search for "bar" instead of "foo".
(replace the searchstring with the selected text)

Ah, I see. Not a bad request