So the whole calling of Jesus to love one's neighbor -- and even one's enemy -- is invalid if someone is not legally right with the government? So no prison ministries then?
It's nice that you quoted scripture, but you really need to quote it with some kind of context:
2 Timothy 3:1-8
This know also, that in the last days perilous times shall come. For men shall be lovers of their own selves, covetous, boasters, proud, blasphemers, disobedient to parents, unthankful, unholy, Without natural affection, trucebreakers, false accusers, incontinent, fierce, despisers of those that are good, Traitors, heady, highminded, lovers of pleasures more than lovers of God; Having a form of godliness, but denying the power thereof: from such turn away. For of this sort are they which creep into houses, and lead captive silly women laden with sins, led away with divers lusts, Ever learning, and never able to come to the knowledge of the truth. Now as Jannes and Jambres withstood Moses, so do these also resist the truth: men of corrupt minds, reprobate concerning the faith.
This passage has nothing to do with whether or not we should love and serve sinners. This is about people who claim to be in Christ, who are putting on an act. Among many traits, they are selfish, proud, disobedient to parents, ungrateful, lack compassion, and false accusers, like many here.
That's very unlike God, for He sends rain upon the just and the unjust. He doesn't withhold common grace, and even extraordinary grace, from those who don't deserve it, which is all of us.
Matthew 5:44-46
But I say unto you, Love your enemies, bless them that curse you, do good to them that hate you, and pray for them which despitefully use you, and persecute you; That ye may be the children of your Father which is in heaven: for he maketh his sun to rise on the evil and on the good, and sendeth rain on the just and on the unjust. For if ye love them which love you, what reward have ye? do not even the publicans the same?
Jesus explicitly teaches a connection and informs us that we will be judged according to how we treat the hungry, thirsty, the stranger (foreigner), the one lacking the basic of life (clothing), the sick, and the imprisoned. You can know whether or not you are a sheep (a disciple of Christ) or a goat (a false disciple of Christ), through the way the gospel affects the way your live toward others.
Matthew 25:31-46
When the Son of man shall come in his glory, and all the holy angels with him, then shall he sit upon the throne of his glory:And before him shall be gathered all nations: and he shall separate them one from another, as a shepherd divideth his sheep from the goats: And he shall set the sheep on his right hand, but the goats on the left.
Then shall the King say unto them on his right hand, Come, ye blessed of my Father, inherit the kingdom prepared for you from the foundation of the world: For I was an hungred, and ye gave me meat: I was thirsty, and ye gave me drink: I was a stranger, and ye took me in: Naked, and ye clothed me: I was sick, and ye visited me: I was in prison, and ye came unto me.
Then shall the righteous answer him, saying, Lord, when saw we thee an hungred, and fed thee? or thirsty, and gave thee drink? When saw we thee a stranger, and took thee in? or naked, and clothed thee? Or when saw we thee sick, or in prison, and came unto thee?
And the King shall answer and say unto them, Verily I say unto you, Inasmuch as ye have done it unto one of the least of these my brethren, ye have done it unto me.
Then shall he say also unto them on the left hand, Depart from me, ye cursed, into everlasting fire, prepared for the devil and his angels:
For I was an hungred, and ye gave me no meat: I was thirsty, and ye gave me no drink: I was a stranger, and ye took me not in: naked, and ye clothed me not: sick, and in prison, and ye visited me not.
Then shall they also answer him, saying, Lord, when saw we thee an hungred, or athirst, or a stranger, or naked, or sick, or in prison, and did not minister unto thee?
Then shall he answer them, saying, Verily I say unto you, Inasmuch as ye did it not to one of the least of these, ye did it not to me.
And these shall go away into everlasting punishment: but the righteous into life eternal.
Serving someone in grace is not "force-feeding" the gospel. Moreover, serving others is not a bribe. That's a complete misunderstanding of the gospel. We serve people as a natural manifestation of the life of Christ in us. If one is truly a faithful disciple of Jesus, one won't consider serving others inside and outside the faith -- even lawbreakers -- as anything unusual at all (see Matthew 25:38-40).
Christians do good and generous things simply because of what God is doing in their lives. It is not about earning or forcing anything, nor about the merit of the person being served. The gospel of Jesus is not a set of doctrine, precepts, and a social code, but an invitation into a life of faithfulness to Jesus. Doctrine, precepts, and perhaps a social code follow discipleship, but it is a result of good things, not the creator of good things.
If one's faith doesn't result in generosity and good works, it is false. Either you have been misled or you are not a true disciple of Jesus.
James 2:14-20
What doth it profit, my brethren, though a man say he hath faith, and have not works? can faith save him?
If a brother or sister be naked, and destitute of daily food, And one of you say unto them, Depart in peace, be ye warmed and filled; notwithstanding ye give them not those things which are needful to the body; what doth it profit?
Even so faith, if it hath not works, is dead, being alone. Yea, a man may say, Thou hast faith, and I have works: shew me thy faith without thy works, and I will shew thee my faith by my works.
Thou believest that there is one God; thou doest well: the devils also believe, and tremble. But wilt thou know, O vain man, that faith without works is dead?