Great questions. Let's talk, and feel free to go back and forth with me on these. And, by the way, conversation about hell is always a worthwhile conversation, because it's so important. The first thing to remember is that the Bible, like law, is true but still must be interpreted. And while people disagree about the interpretation, just as they do about the law, that doesn't make the law or the Bible untrue. Lawyers and Christians should feel free to discuss, debate, and even disagree. After all, the most important things are worthy of our best thoughts, and there are bound to be disagreements. It's OK.
You are right that hell is first and foremost a place of separation from God, and therefore from Life (God is life, eternal and self-sustaining). The Bible often uses images of fire to describe it, but lots of times it also does
not use images of fire to describe it. The Bible uses five main pictures to speak of hell:
1. Darkness (Mt. 8.12; Jude 13) and separation (Lk. 13.27-28; 2 Thes. 1.7-9)
2. Suffering and Remorse
* Weeping and gnashing of teeth (Mt. 8.12; 22.13; 25.30; Lk. 13.28)
* cut to pieces (Mt. 24.51)
3. Punishment (Mt. 25.46; Rev. 14.11)
4. Fire (Jude 7; Luke 16.24)
5. Death and destruction (2 Thes. 1.7-9; Rev. 20.14)
I happen to think hell is not literally fire, but the agony of true separation from God. I say that because fire doesn't have degrees of punishment, but hell does. Degrees of separation makes more sense to me than degrees of being burned. I believe hell is degrees of punishment, based on the sin (though not levels of hell, as in Dante). Here's my proof:
Mt. 11.22-24 – “more tolerable”
Mt. 23.14 – “greater condemnation”
Rev. 20.13 – “each in proportion to his works”
Lk. 10.12 – “it will be more bearable for Sodom than for that town”
Lk. 12.47-48 – beaten with few blows or more blows
But now to your specific questions. What is the second death? It's mentioned only four times in the Bible: Rev. 2.11; 20.6, 14; 21.8—obviously all in Revelation. Revelation explains it as "the lake of fire". It is referred to as the second death, because people have already experienced the first death—separation from physical life. The lake of fire is referred to as the second death—separation from spiritual life, or complete separation from God. The second death is for those who will share the eternal fate of Satan, the beast, and the false prophet (comprising the anti-Trinity).
Is the second death a reference to annihilation? Very few believers accept annihilation. The teachings of the Bible are very strong against it, though some Christians do believe in annihilation. 2 Thes. 1.9 is a strong text against it, though there are many.
Is hell eternal? Again, 2 Thes. 1.9 is a strong statement on the eternity of hell, but you should be aware that there are great efforts to understand the teaching of the Bible about eternal punishment. There are certainly verses that talk about eternal punishment, but they don't necessarily include all of those who are separated from God. There are verses that talk about God reconciling all things to himself (Rom. 11.15; 2 Cor. 5.19; Col. 1.20), and so some theologians think that God will continue his work of reconciliation even into "eternity", such that those who "serve their time" will at a later date be reconciled with God. There is another position called "semi-restorationism" where, after appropriate punishment, those who desire a relationship with God will be partially restored, and those who do not, even after punishment, will opt to remain separated. The point is that, consistent with everything else the Bible teaches, a person's relegation to hell will be something the person will choose ("I don't WANT to be with God!") rather than a destination forced upon one against their will. While the Bible speaks about eternity, possibly only those who stay eternally defiant will be eternally punished. It's hard to know. The language of the Bible speaks over and over about hell being eternal, but some theologians think that "eternal" is able to be interpreted and may not be what it seems to sound like on the surface.
Is the second death something that takes place in Hell, or is it a different step in the process? It seems to be the last step. In Luke there's a story of a rich man who is burning in hell, in torment. But then in Rev. 20.14, death and Hades are thrown into the lake of fire. What is usually understood is that hell is the place unbelievers go after death, but then after the final judgment, there is another step of separation from God, the final one, which is eternal destiny (depending on how one interprets "eternal").
My bottom line is this: Those who turn away from God will be separated from the life of God. Though we can't be sure about the form or duration of that separation, this we can be sure of: it will be a horrible experience, and God will be fair about the form and duration of it. If you reject God, you take your chances.
You want to know what the proper position is to take. A few Christians believe in annihilationism, a few in reconcilationism, a few in semi-restorationism, and the majority believe in the traditional view (which is why it is the traditional view). To read Scripture in the simplest and most obvious way, hell is the eternal destiny of the unsaved. Many people wrestle with the justice of that, and so try to come up with other ways to juxtapose God's love, his justice, and hell, and that's where the other theories come from. These are worthy pursuits. I think eternal, conscious hell has far stronger Biblical support than annihilationism, but such questions shouldn't decide whether you believe in God or not. Study the Scripture yourself, and see what the Lord teaches you. As for me, I believe in what I wrote that I called "my bottom line". the form and duration of hell may be "negotiable," but God's justice is not. That's what I hang my hat on.