All of this depends on if the original creator of the joke is the person who originally tweeted it. On Twitter, there is an implied, if not explicitly stated, permission to re-use and recombine tweets. However, if the creator of the joke did not tweet it, then those who do tweet it may be violating copyright. Yes, everything you create is copyrighted as soon as you record it in some way. If others use that without your permission, either explicit or implied, then they are violating your copyright. Just because Twitter is based on copying doesn't mean it is inappropriate for them to protect the copyrights of works placed in their system by people other than the original creator.
Really, the simplistic view of the world taken by supposedly smart people on this site never fails to astound me.