Yes, the young man was not saved, because he lacked faith in Jesus Christ as Savior. He thought he was perfect and could go to heaven because he was. When Jesus pointed out his weakness - his wealth - he showed him that his perceived perfection was never going to be enough. Note also that Jesus didn't say to him that if he gave everything to the poor and followed him, he would be saved, but that "his reward would be great in heaven". There is a difference. The base of our faith is that Jesus Christ is the one and only Savior and we should not allow anything in our lives - family, love, money, fame - to keep us from following him.
What do you mean by this? Are you saying that unsaved people can get to Heaven? That goes against the Scriptures which tell us that nothing imperfect can enter the Kingdom of Heaven. Or are you saying that this unsaved man will have rewards in Heaven apart from his eternal Judgement to hell? Nothing from Heaven passes to hell - remember that the damned man was denied even a drop of water touched onto his tongue. What exactly are you trying to say?
No, that is not why he was not saved. Jesus explained why:
Yet one thing is wanting to thee: sell all whatever thou hast, and give to the poor, and thou shalt have treasure in heaven: and come, follow me (Luke 18:22)
Why don't you read what is written? He was wanting "one more thing". That one more thing was not faith, as you say, it was giving away his possessions. Further, there is no evidence that he lacked faith: he asked Jesus, "Good master, what shall I do to possess everlasting life?" (Luke 18:18). One does not ask a question unless he has faith in the person giving the answer.
that Jesus didn't say to him that if he gave everything to the poor and followed him, he would be saved, but that "his reward would be great in heaven".
That is spin way off the page. "Heaven" is synonymous with "evelasting life". The question was about "everlasting life". Do you think that "saved" has a different meaning as "have everlasting life in heaven?"
we should not allow anything in our lives - family, love, money, fame - to keep us from following him.
That is of course true, but the story tells us that a specific work of charity was required of the young man. The unwillingness to do that work of charity is what prevented him from gaining eternal life. It is more that some abstract "not allowing to keeps us from Christ" it was actively doing certain work.
Read what is written.