The Wayback Machine is actually a holodeck.This explains everything about it: the cartoonish portrayals of history, the fact that nothing Peabody and Sherman do seems to actually matter, and even the Translator Microbes. Peabody just wants Sherman to think it's a time machine because he [Peabody] thinks it'll seem more interesting that way.
The real reason for Peabody's bad puns at the end of each of his segments.They're the only way to tell the Wayback machine to turn off; the saying of the bad pun returns them to the time they departed from. Trying to figure out why he built the machine this way seems to me like trying to figure out why the Garbageman is a garbageman even though he's the world's smartest human; because we are not Peabody, we will never truly find out.
Bullwinkle is in fact not a mooseHe is a were moose, he is human for half of his life that's why he went to Whatsamatta U., a moose could never be accepted into College.
Rocky is castrated.Ever wonder why his voice doesn't break, gets taller, or grows facial hair for 52 years?
Peabody is a Time LordThe Wayback Machine happens to be his TARDIS. As for how a Time Lord became a dog, probably just a weird regeneration.