I'm not sure I understand your second question. Maybe the reason it's reflexive is that if you wish for something, you want it yourself. If you wish for someone else to get something it would be syntactically very similar but you'd use a different object. (å ønske noen noe) Here there is probably only a reflexive pronoun because you need an object.
my wife wishes a younger man is not a complete sentence in English. If you want to use wish, you'd have to say something like My wife wishes she had a younger man or or My wife wishes that she had younger man or My wife wishes for a younger man (even that sounds odd in English).
Tough audience, down-voting you for asking a good question.
The style of the sentence you've suggested would have worked quite nicely about 400 years ago. In seventeenth-century English, it was grammatical. Now, it's just too old-fashioned sounding. You can substitute a couple of different verbs for your sentence, and it still works fine in English:
My wife desires a younger man.
My wife wants a younger man.
You could say, My wife wishes to have a younger man (very formal)
My wife wishes for a younger man (even more formal--use this only if you're the butler of Downton Abbey making a confession to the parish priest).
Your last line made me laugh but I don't really agree with it.
If someone makes a wish whilst blowing out their birthday candles (is that a normal thing everywhere?) what you ask them is "What did you wish for?" and it would be perfectly normal to reply "I wished for a younger man."
Well, it wouldn't be perfectly normal because, of course, the correct response is "It's a secret, otherwise it won't come true." But in terms of the sound of the sentence it's perfectly normal.
I'd say using "desires" sounds much more formal.