I disagreeingly agree with you. Also in English you can use 'beg' instead of 'ask' (as in "I beg your pardon"), still in most circumstances it's considered archaic (you wouldn't say "Please, may I beg for a tissue?"), and nowadays you use "beg" in a sense which would translate to "betteln" into German https://de.wiktionary.org/wiki/betteln
The short answer is "Usually, but not this time."
You can in principle combine verbs as insanely as you want. You can say "Ich will klavierspielenlernen" if you want to, and it's only a bit weird. (But much better "Ich will Klavier spielen lernen".) If it's a noun, this isn't even bizarre, just clunky: "Ich mag das Klavierspielenlernen". Even "Ich habe das Klavierspielenlernenwollen" is grammatical and comprehensible, though the best I can say in its defense is that it's not technically illegal, and you can go further...
But the compound that you suggest in particular doesn't mean what you want it to. When you add a new part to a verb, it acts sort of like a prefix, so in your word "bitten" would be a prefix of "gehen"; in the sentence, "Ich gehe euch bitten", which is not what you want. And you can't get what you want by switching them around because you can never get the zu in there - it's not a prefix of anything and can never become one, even if you're so loose about prefixes that you'll consider "Klavier" one above.
"Verlassen" means "leave" in the sense of "abandon" or "desert" (or the specific context of "leaving someone", i.e. terminating a romantic relationship). So it requires an object; you have to verlassen something. But even with objects added, asking someone politely to abandon something or leave someone is a bit strange.
Probably not; try breaking up the sentence into parts: "Ich werde euch bitten", "zu", and "gehen". Part 1 and Part 3 are connected by Part 2, just like in English "I would request you", "to", "leave". I don't suppose this is the best way to explain it, but I hope it helps you understand it better.
It's infinitives that move to the end, not all verbs. ("Werde" is in second position where conjugated verbs belong.)
And the infinitive moves to the end of the clause, not to the end of the entire sentence. "Zu gehen" is a new subordinate clause, so "bitten" goes before it.
Note that it could be extremely confusing if infinitives moved to the end of the sentence, especially if there was a longer subordinating clause in the sentence. "Ich werde euch bitten zu gehen, weil ihr zu laut seid" is understandable, but moving "bitten" to the end-- "Ich werde euch zu gehen, weil ihr zu laut seid bitten"-- takes a lot of work to figure out where that "bitten" is coming from. So "bitten" just goes to the end of the clause.
Based on a couple of dictionaries I checked, "bid" means ordering or instructing someone to do something. "Bitten" is just a request.
Either way, though, don't expect Duo to have old-fashioned expressions like that; it never does, since those just aren't useful translations for the modern day.