OK. Gary Fisher basically invented the 'mountain bike' by bolting motorcycle parts on to a bicycle frame for the increased strength and performance he needed for his downhill habits.
Of course, you can use bicycle parts. it's a question of what, where, how and why.
If you just want to save money, there are a lot of crappy bike parts. Maxxis is a new name for Cheng Shin, which in my experience have been some of the crappiest moped tires in existence. However, they make some nice motorcycle and bicycle tires.
I have bicycle brake cables and derailleur cables on my Moby, albeit high quality ones, actuating everything. I have motorcycle grips, though. Bicycle grips don't suit my needs.
You can't put a derailleur on the engine driven chain of a moped without some serious engineering - all of which is easily trumped by a variator or transmission, which is probably why nobody has bothered.
Wheels should be considered for their designed strength - I would not trust most bicycle wheels to what my moped would put them through.
That said, a properly built wheel of suitable girth to give me confidence could, theoretically, be a bicycle wheel by design, but I haven't seen one yet.
In short, generalizing will get you nowhere. Post up the rims / parts you're considering, with some real specifics about them, and specifics about your bike, and then we can talk intelligently.