> Should I rename that app too?

Yes, in my semi-informed, non-lawyer opinion, you should change the name and remove all references to the FB Messenger icon. I don't think Facebook has a word mark on "Messenger", from a quick USPTO search, but using the name makes it harder to defend your use of trademarks like this one within the app: http://tmsearch.uspto.gov/bin/showfield?f=doc&state=4806:9la...