Everybody puts his hands in the air or Everybody put their hand in the air. The difference here is that "his" vs. "their" mean the same thing, but differ in gender specification. Also, if "hand" or "hands" depends on how many hands EACH person puts in the air, not on how many people are doing it. Everyone jump in the muddy puddle or Everyone jumps in the muddy puddle. The difference between these two, for me, is that the first one is a command and the second one is a description/narration.

indefinite pronouns, like everyone, anybody, anyone, everybody etc may seem to have plural meanings, but they are singular in formal English. So " Everyone jumps"

If you say "Everyone jump in the muddy puddle" it sounds like a demand. It can be like saying, everyone here, jump into the puddle now. If you say "Everyone jumps into the muddy puddle" people will read this as a description of a present tense event. It would be like saying, everyone is jumping into the muddy puddle.

As to the part of your question when you say his or their, in english when we speak about someone in third person and we don't know his or her gender, we always use 'their'. Their is used in both singular and plural.

When you say everyone, or everybody, it is really a third person singular. The clue is in the part of the word 'body' or 'one'. So if you are trying to work out the corresponding verb, think about it like this: Construct the sentence and instead of saying Everybody or Everyone, say 'Joe'. Eg: Joe jumps in the muddy puddle. This is describing the the action of Joe. Now just swap Joe for Everybody, or Somebody, or Nobody! Let me know if you need more explanation :)