Why not?
You can be a murderer and a Christian. You can be a thief and a Christian. You can be a liar and a Christian, you can be an adulterer and a Christian. All those things are sins, and sure, a Christian should shun sin and fight against the temptations of this world. But we all fail, and we all fall. That is, after all, why we need a saviour.
Yes and no. Look at the tenses of the relevant Greek in I John, especially chapter 2. We all sin, true, but all the same a person cannot be habitual in some sin, they cannot continue in some sin without remorse and without repentance, and call his or herself a Christian. Yes, a Christian may lie or steal through weakness, but a person who habitually lies without feeling any need to repent of it and get right with God (and then doing so) cannot BIBLICALLY claim to be a Christian. A person who makes a lifestyle out of stealing things from other people and continuing in that sin cannot rightly claim to be a Christian. Likewise, homosexuality - which is an ongoing lifestyle - basically proves a person to not be a Christian. Remember, by their FRUIT ye shall know them - fruit is something that develops and grows over time. If grace has to be stretched to cover ongoing, habitual, unrepented of sin, then no grace has been had in the first place.
Without repentance, there is no salvation (Acts 3:19).