Yes, it certainly does.
I like your first sentence, but I would change “behave” to “believe.” If the individual believes as a Christian ought to, the behavior will follow. Good works are certainly the fruit of a saving faith, and are a wonderful proof of the true Christian’s rebirth. Your second question seemed to anticipate that answer.

But I think it mistakenly places the focus on the created rather than the Creator. The question isn’t whether one must do good works in order to be saved (which you may be right to identify as selfish), because God doesn’t need our good works. To Him, our good works are about as valuable as a rusty tin can (someone can find the Bo Giertz reference for me). But our neighbors need those good works. And so we follow that second Great Commandment Rabbi Hillel understood from the Law and the Prophets and that Jesus confirmed: love your neighbor as yourself.
The dominant desire of a Christian is not so much to be saved, but to conform to God’s Will in thanks for the gifts he has received. God’s Will is for all to be saved and come to the full knowledge of the Truth. So in that regard, the desire to be saved - to return to communion with God - is not sinful. It is the desire of a repentant heart.
I’m sorry if you feel I haven’t answered those questions to your satisfaction. They were posed in a way that placed the impetus on man to save himself. That’s a paradigm that I don’t see supported in Scripture.