Zipping right off to heaven or hell is modern popular theology, but it is a reinvigorated one as well; but not a good reinvigoration. The reason is because it is something Gnostics, not Christians, taught due to their belief in the superiority or reality of the immaterial over the inferiority or illusion of the material.
Besides, Christianity makes it very clear that it teaches a Judgment at the end of temporal time as we know it, in which we will then be determined worthy to experience heaven or hell, and it is equally made clear that both the living and the resurrected dead will be Judged together, so unless we wish to accept full preterism (which is a heresy and historically inaccurate since the Second Advent has yet to occur), it is also unbiblical and illogical.
Instead, the Bible teaches that we shall dwell in sheol, called hades in Greek, which according to extracanonical and canonical texts is divided into two "departments": Paradise, also known as Abraham's Bosom, for those who are given particular judgment as righteous, and gehenna for those who are given particular judgment as unrighteous.
Soul sleep/death is also a foreign Pagan concept. Souls are not eternal, but they are immortal, for they are life. We are not however truly a person without our body, which is just as important as our soul, and the separation of the two, which we commonly call "death", is the result of Adam and Eve's initial sin, which is our fallen state; this is the doctrine of Original Sin. The body returns to the earth but the soul waits. And since a soul is life and immaterial, a soul is literally tireless; it has no form to prohibit it from exercising its potential. So, naturally, the soul does what all souls do that wait: souls pray. Souls pray for themselves, they pray for those still living their temporal lives, and they pray for the hastening of Jesus' Second Advent.
Again, that is actually a Gnostic idea, not a Christian one.
Not really. Consider what Jesus said of it in His Parable of Dives and Lazarus. It isn't just a flowery story; it illustrates proven Jewish and Christian belief about the immediate afterlife.