Whether content from Facebook can be used as evidence is a hotly debated topic in the current legal landscape.
You should consult with a lawyer in your state to discuss how Arizona's rules of evidence treat social media. Admissibility is a very complicated issue, and I don't recommend trying to figure it out on your own.
If it can be authenticated, probably yes - but you should contact a local attorney familiar with the rulings of your judge to be more certain.
Under the rules governing the conduct of attorneys in New York it may be necessary to remind you that this answer could be considered attorney advertising.
All information regarding the fitness of a parent is something that can be presented to the court. Evidence to support that is something you want to discuss with your attorney.
If you have a lawyer, ask them; if you don't - the worst thing the judge can likely do is not allow the matter into court, so there is no harm in trying from that end. I'd be a bit concerned about admitting under oath that you used his password to gain access to an account belonging to him without his permission, as I don't know if you'd be opening another can of worms.
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