1. Not necessarily. The Addict could come to Christ and make him his/her Savior, King, and Lord....yet still have this addiction which came along with him ; but when a new Christian gets the power of the Holy Spirit dwelling in him/her, they have the power to overcome an addiction . God wants us all to be Overcomers of things that harm our Walk and Relationship with him., and he is only too willing to help us. It also requires OUR cooperation by wanting rid of the addiction (repenting) so we don't continue to be defeated by something that is not of God. As we allow God to quicken our Heart, we get supernatural power and zeal to come against anything bad in life . We come to Christ UNCLEAN, and he washes us and cleans us up . But we have to want that also.
2. Falling back into the same addiction after a time, is common with many People who are truly saved ; its because we are weak in our own nature and strength which is why we truly need God working in our lives and Souls. Some addictions require going to a Professional Counsellor whos had experience with the addiction ...other addictions can be handled relying on God and attending workshops like Celebrate Recovery , a national organization in all major cities , and also thru the soliciting of prayers and support from trusted Friends. God is patient with us when we slip away and he knows whether the slipping away is willful on our part or if we are just simply powerless . As fellow Christians, we should be there to help support the true Person who wants to beat the addiction .
3, The Person who WILLFULLY turns back into a lifestyle sin addiction and who has confessed Christ , is an unsaved Person according to Gods Word in many places. A genuine submission and sorrow for ones sins when recieivng Christs free undeserved offer of forgiveness is different from a temporary moment of wanting Christ only to soon go back to the lifestyle of sin because you missed it and because it has become your true 'master' . Whatever is truly important to you, there your heart shall be also. A Person cannot serve both the World and God at the same time, for, he will hate one (ultimately) and serve the other.
The key to falling away temporarily, is to keep stayed on God and his love and promises and power to bring you out of it and to make it permanent . Christ came so we would be Overcomers of sin ...so sin wouldn't have power over us and keep us pinned down in enslavement regardless of how good it feels at the time.