This is a particularly acute problem with Buddhism because an explanation of the religion does not translate well into the English language. For example:
The first noble truth is that suffering is an unavoidable and basic condition of human existence.
This sounds very pessimistic, doesn't it? However, what it actually means is that no one can live a perfect life free of hardship and pain. Even if we have wonderfully good times, we still "suffer" when those good times come to an end. I've always thought suffering is a bad translation because it implies a tremendous amount of pain whereas Buddhist "suffering" can be as simple as a small disappointment.
The same is true of the concept of "nirvana." I've read it described as annihilation, which seems like a silly goal to work towards. It does not really have a good English analogue.
This thing you posted is an example of what us Buddhists fear this "outreach" will be. I highly doubt this was written by a Buddhist evangelical as it has quite a few outright lies and misinterpretations.
There is no God.
That is simply not the Buddhist view. The Buddha was completely silent on the concept of God. He viewed the greater workings of the universe to be irrelevant to the individual's path to enlightenment. Some Buddhist schools believe in a God, some deny he exists, some have many Gods. Buddhism has historically combined readily with indigenous beliefs it has come across as it spreads which is why a lot of people talk of Buddhist Christians.
Train oneself to acquiesce to extinction without remainder. So, for example, if you happen to trip while going down stairs, you can just give up in midair and not try to save oneself and increase the likelihood of dying without the possibility of rebirth.
I'm not really sure where to even start here, because this is all wrong. The goal of Buddhism is not to just give up and cease to exist. Buddhists want to live their lives to the fullest just like everyone else! Enlightenment is not achieved through nihilism.
Seeking after this acquiescence for the purpose of ending one's suffering is itself an example of grasping desire that keeps the whole thing going. Although advanced meditative techniques can be used to get around this, simply training oneself to give up when faced with life-threatening situations can be helpful because the habit of acquiescence can help you to encounter death without the desire to stay alive and, thus, break the chain of desire and existence.
Same here. The goal of Buddhism is not death! It is not a suicide cult!!!!
Sorry if this has gone on too long but I hope this helps you better understand the worries of Buddhists when it comes to conversion, especially by Westerners. The concepts are very alien.