What doctrines are based on 1st Timothy 4:14 when hands are laid on a believer?

As ever, context is all important when trying to understand this verse. The gift in question is not the laying on of hands itself, but Timothy's ability to teach and preach:

1 Timothy 4:11-14 (New International Version):

Command and teach these things. Don’t let anyone look down on you because you are young, but set an example for the believers in speech, in conduct, in love, in faith and in purity. Until I come, devote yourself to the public reading of Scripture, to preaching and to teaching. Do not neglect your gift, which was given you through prophecy when the body of elders laid their hands on you.

The Message puts it even more clearly:

Get the word out. Teach all these things. And don't let anyone put you down because you're young. Teach believers with your life: by word, by demeanor, by love, by faith, by integrity. Stay at your post reading Scripture, giving counsel, teaching. And that special gift of ministry you were given when the leaders of the church laid hands on you and prayed—keep that dusted off and in use.

So to answer your last question first, references to the same gift given to Timothy (a teaching/preaching ministry) can be found in many places in the Bible. I won't list them here as any decent concordance (or a search of an online Bible) can do that for you.

But the main thrust of your question concerns the laying on of hands. The main principle behind this is a channelling of (for want of a better phrase) spiritual power through touch. Often in new churches you can see someone with one hand raised in the air (towards God) and the other laying on someone's head, shoulder or back while they pray for them - symbolising that the person praying is acting as a conduit, a link between God and the person being prayed for.

As well as the laying on of hands to confer a prophecy as in this case, it's often used in prayers of healing. There are numerous examples in the gospels of Jesus healing people by touching them physically:

Luke 8:43-56 (MSG):

In the crowd that day there was a woman who for twelve years had been afflicted with hemorrhages. She had spent every penny she had on doctors but not one had been able to help her. She slipped in from behind and touched the edge of Jesus' robe. At that very moment her hemorrhaging stopped. Jesus said, "Who touched me?"

When no one stepped forward, Peter said, "But Master, we've got crowds of people on our hands. Dozens have touched you."

Jesus insisted, "Someone touched me. I felt power discharging from me."

When the woman realized that she couldn't remain hidden, she knelt trembling before him. In front of all the people, she blurted out her story—why she touched him and how at that same moment she was healed.

Jesus said, "Daughter, you took a risk trusting me, and now you're healed and whole. Live well, live blessed!"

While he was still talking, someone from the leader's house came up and told him, "Your daughter died. No need now to bother the Teacher."

Jesus overheard and said, "Don't be upset. Just trust me and everything will be all right." Going into the house, he wouldn't let anyone enter with him except Peter, John, James, and the child's parents.

Everyone was crying and carrying on over her. Jesus said, "Don't cry. She didn't die; she's sleeping." They laughed at him. They knew she was dead.

Then Jesus, gripping her hand, called, "My dear child, get up." She was up in an instant, up and breathing again! He told them to give her something to eat. Her parents were ecstatic, but Jesus warned them to keep quiet. "Don't tell a soul what happened in this room."

The laying on of hands is not essential for God to bless the recipient (as shown when Jesus heal's a woman's daughter without the daughter being present, as outlined in Matthew 15:21-28) but the passage above shows how powerful and effective physical touch can be,and the laying on of hands is simply an expression/extension of that.