Without an annulment, a convert is confirmed and receives first communion A non-baptized man marries a Catholic, then divorces.  Then the non-baptized man remarries a Protestant.  During this marriage the man is baptized Protestant. The man later coverts to Catholicism and but does not get an annulment from the first marriage because the priest says he meets all three criterion and absolves him. 

He is then confirmed and receives first communion, from the same Msgr.   


Is that man able to receive first communion and confirmation?
Is he in a state of grace?
Can he be a godfather?
"Marriage possesses the favor of law; therefore, in a case of doubt, the validity of a marriage must be upheld until the contrary is proven"

I refer to the second, current marriage.  

Your question is both incomplete and nearly incoherent. You mention "all three criteria" yet you do not spell out what you believe them to be. 

It is quite possible that the first marriage "lacked form" and thus in the eyes of the church never was.   


Caveat: in my experience with RCIA, that situation you describe usually led to a decree of nullity (the short form) if a divorce decree was in the petitioner's possession, and, if the original spouse, even though Catholic, had not married in the Church with this non-Catholic person.      

It is the responsibility of the Catholic party to do that for the marriage to be a sacrament.  If she had married outside of the Church in his first marriage, it is really her fault (in the eyes of the Church), not his, that the first marriage was not sacramental.  I have seen multiple variations on this, so it requires a lot more information than you provided to come to a finding.  I will assume good faith on the part of the Msgr in this case and estimate that such an inquiry took place. If it did not, then someone didn't do their job.  Action item 1: Go and talk to the Msgr and find out what happened. The internet can't help you.        


Existing marriages have the favor of law

Even if such a decree of nullity was not issued for the first marriage, his second seems to be in good order.  Since the Church typically holds that a marriage has the favor of law unless it is challenged, and, the Church generally holds the marriage of other Christian couples are licit and valid (unless challenged) - there doesn't seem to be a problem beyond (perhaps) a paperwork problem in that diocese.     


  Can.  1060 Marriage possesses the favor of law; therefore, in a case of doubt, the validity of a marriage must be upheld until the contrary is proven.  


If the second spouse is still this person's spouse, there may be no need to raise any challenge.     

Incoherence and Incompleteness

This is what you say:   


  He is then confirmed and receives first communion, from the same Msgr. 


You then ask:   


  Is that man able to receive first communion and confirmation?   


He already did, so this question makes no sense.     


  Is he in a state of grace?  


At the moment of receiving communion, yes. After that, who knows?  We can fall into sin as soon as we walk out of the door of the church.  (Hopefully, we don't)  


  Can he be a godfather?  


Maybe, probably not for a Catholic baptism.  That is a separate question.  The requirements of being a godparent are a bit more than "received communion once."  Consult with the parish priest and, if ncecessary, the chancery at the local diocese.  

Caveat:  If the second spouse isn't Catholic, this will take a ruling from the diocese for this to be favorable, since the usual requirement is a Catholic married couple. (In my experience with RCIA in a US-based diocese).  

God parents for baptism, and requirements for that role, are best inquired about at the local parish where the baptism is being performed.  Here are the requirements in the Dioces of Austin, Texas.  


  REQUIREMENTS FOR GODPARENTS FOR YOUR CHILD
  As stated in Canon 874 and required by the Diocese of Austin
   - Must be at least 16 years of age.
   - Must be a baptized Catholic who has completed the sacraments of Eucharist and Confirmation.
   - May not be the parent of the child being baptized.
   - If married, must be married in the Catholic Church, regularly attending mass on Sunday, and living their Faith.
   - The two godparents do not have to be married to one another. 
   - A baptized non-catholic may not be a godparent but may serve as a witness along with a Catholic godparent.
   - Non-baptized persons may not serve as witnesses or godparent   


Short answer: likely can't be a godparent without a special dispensation from the Diocese.  

You seem to be asking if the Msgr did the right thing.

What your question seems to be is asking whether or not the Msgr made a mistake, and that can't be answered with the information you gave.  

The most likely thing that happened was that, during the conversion process (usually RCIA in a case like this) the marriage history was investigated and a series of 'yes/no' questions were answered per the guidance given by the diocese.  If the Msgr received 'yes' answers, then he was confident that all of the Canon Law requirements were met, and confirmation of the Candidate was completed per the diocesan guidance.  

If the Msgr wasn't that thorough, then there may be a problem that can only be resolved in the diocese where this happened.   Once again, contact the Msgr and find out the details. The Internet Can't Help You.