Learn English – Antonym of the verb “complete”

In this case, I think "would" is being used as the subjunctive of "will," "will" meaning to desire. So in the case you describe, it means something akin to "wish." I admit it sounds archaic, but I'm pretty sure it's okay. I know Shakespeare uses "I would it were..." often enough. And as you set out in the question, we still use it today.

The first two sentences, as written, don't make sense. "Rather" is not a verb. I've never heard it used like one, either. I'm guessing if you've heard that it was a mistake, and they meant to say/ type "I'd rather..."

So, as far as I know "I rather my opponents don't find out" doesn't make sense. "I'd rather my opponents don't find out" does. You can insert a verb into most of the sentences if using "would" as the subjunctive of will bothers you, but it's not necessary.

Usually you can just slap re on the front of a word, without a hyphen, and you will be understood. There’s no need for the term to have made its way into an “authoritative” dictionary, even if there were such a thing.

There are circumstances in which the hyphen is desirable:

• if your new word with re collides with an established “re” word which has a different sense. Two of the words Carlo_R mentions are of this sort: resign and reserve.
• If you want to draw attention to the fact that you are using a “re” word in a different sense than that which would be expected, particularly if you are reverting to the base sense. “D.W.Griffith’s re-construction of the myth of the carpetbagger.”
• If you want to emphasize that you are repeating or reversing an action. “Burns opened an interesting question in 1923; but it has been completely neglected and I want to re-open it now.”

I sometimes intrude a hyphen when the word to which I’m attaching re begins with a vowel, to prevent even momentary confusion; but it’s probably not necessary.