One's sexuality is irrelevant to one's salvation. Jesus came to save everyone, both straight and gay.
Of course. But Jesus also makes it clear that those who do not follow His commands, those who do not remain in Him are not following Him.
If you believe Jesus lived and died for you, you're a Christian.
Belief is not enough. That much is made clear in the Bible. Even the demons believe:
Jas 2:17 So faith by itself, if it has no works, is dead.
Jas 2:18 But someone will say, "You have faith and I have works." Show me your faith apart from your works, and I by my works will show you my faith.
Jas 2:19 You believe that God is one; you do well. Even the demons believe--and shudder.
Jas 2:20 Do you want to be shown, you senseless person, that faith apart from works is barren?
I'm not an expert on what constitutes a Christian, not least because there are several major and conflicting schools of thought, but that seems to be the key point. Everyone's a sinner, everyone can be saved. Even Christians sin, be that sin theft, lying, lusting, coveting, or even being a Big Gay™.
Everyone can be saved if they repent of whatever sin they may be living in and decide to follow Christ:
Rom 6:1 What then are we to say? Should we continue in sin in order that grace may abound?
Rom 6:2 By no means! How can we who died to sin go on living in it?
Rom 6:3 Do you not know that all of us who have been baptized into Christ Jesus were baptized into his death?
Rom 6:4 Therefore we have been buried with him by baptism into death, so that, just as Christ was raised from the dead by the glory of the Father, so we too might walk in newness of life.
Rom 6:5 For if we have been united with him in a death like his, we will certainly be united with him in a resurrection like his.
Rom 6:6 We know that our old self was crucified with him so that the body of sin might be destroyed, and we might no longer be enslaved to sin.
Rom 6:7 For whoever has died is freed from sin.