Forget that he is a recovering addict for a minute, because you are probably cutting him too much slack for this reason. Should he have carte blanche because he is in recovery? and if so, for how long precisely?
Look at him as your partner and his behaviour, full stop. Is that behaviour making you happy or miserable and is it likely to make you happy or miserable in the future?
Very snappy at times. Red flag.
Very quick to get loud when your tone is normal. Red flag.
Thinks your trying to attack him ie defensive (paranoid?). Red flag.
Small (unneccassary) lie. Red flag.
Making excuses for said lies. Red flag.
The lying is a big thing. It lacks integrity which will come back and hurt you time and time again. If there are little lies you can be sure that there is or there will be a big lie. Compulsive liars don't know where they stand on anything because they distort the truth so much. They lack authenticity and a relationship with such a person will gaurantee a loss of security. Some will tell you what they think you want to hear...
He is willing to go talk to somebody about it. Good for him. Do you want somebody with baggage and a wait and see if one day this resolves kind of feeling? Or a partner with their house already in order?
Some have said you are being very supportive but you may want to ask yourself if you are being in fact, a doormat.
Look at the situation as if it was happening to your best friend and her partner...what would you advise her to do, to best take care of herself?
By the way, cheaters often send flowers "from out of nowhere". Just a thought.