There are two times where a character sacrifices themselves to a blast that would be lethal to protect someone else, but because of how the game works they come back later on. There is a character who nearly let them themself die before being grabbed and carried out of a room that was going to explode though.
From what I know, there is the Bubble Baby Mario sticker which is based on the same occurrence that can happen to him in the Yoshi's Island games (when Yoshi gets hit, which causes him to lose/ drop Baby Mario who is stuck in a bubble and cries until Yoshi gets him back) but aside from that, there's no audible crying babies in game
No, but a couple of characters came very close to it in the storymode. In gameplay though, the main way to beat your opponent is to knock them off the stage which could lead to something similar.
A character, who is a giant hand, was stabbed in the knuckles and stuck with what stabbed them until they got ripped out, with visible (but not human colored) blood. That scene happens pretty quickly, but afterwards there are burn marks left where they were stabbed.