I have a very hard time keeping myself pure, avoiding lustful thoughts. It seems like every minute there is a thought I have to sweep away. I have shared this with my wife and fellow disciples to get it into the light, but it still happens. This brings me to the beatitudes, specifically Matthew 5:8 ("Blessed are the pure in heart, for they will see God"). I strongly believe that I will not see God, and this saddens me. I believe that I might make it to heaven, but won't be allowed to see Him. This has made me cry. Am I right about this?

No, I do not believe you are right about this. If you die saved, you will see him (1 John 3). God will not avoid us, as David avoided the sight of Absalom (2 Samuel 14:24). 

Besides, the beatitudes speak about the heart, about the attitudes. Purity in Matthew 5:8 may refer to sexual purity, but I doubt it. I think its scope is much broader. Jesus is really talking about what drives us. Are we selfish, or selfless? Are we dishonest, or honest? Are we devious, or straightforward? Do we strive to exalt ourselves, or to exalt Jesus Christ?

So, while it is always good to develop more self-control (and also rely on God's strengthening grace to do so), "pure in heart" refers to motivation more than to right actions. Not that they're unconnected. But the fact that your sin deeply bothers you leads me to believe you probably are pure in heart. Please think about this.

Suggested reading: Tom Jones, Strong in the Grace; Sam Laing, The Guilty Soul's Guide to Grace; Philip Yancey, What's So Amazing About Grace; and many other helpful volumes.