The only rule requirement is that the wrestlers shake hands before the match starts and at the end of the match as well. Failure to do so would be unsportsmanlike conduct.

Shaking the opposing coaches' hand is a Kansas custom, at least in most places, but failing to do so only raises an issue between the wrestler and his coach. I believe Missouri actually discourages the practice so it is not a sign of disrespect when they do not shake the opposing coaches' hand. It does cause a little confusion sometimes.