Okay, so to understand the interrelation, let's start by examining the situation where all three conditions apply. Note: I am intentionally omitting diagrams. Partially this is to give you a chance to work on your visualization skills, but if you have any difficulty understanding any portion of this post, by all means set it up on a board and work through it. It's also partially to work on MY visualization skills by attempting to explain all of this without a board in front of me. So I would appreciate if someone would double-check that I didn't say anything stupid
White has his king in front of the pawn, and on the 6th rank, with the opposition; for example the white king is on e6, pawn on e5, and the black king is on e8 - black to move. Hopefully this is clearly a trivial win for white. Because white has the oppposition, the black king is unable to advance - all three squares within its reach on the seventh rank are covered by the white king. Since black cannot bring his king to d7, e7, or f7, he must instead sidestep to either d8 or f8. Whichever direction black moves, the white king can come forward to the seventh rank on the other side and take control of the queening square (1. ... Kd8 2. Kf7 or 1. ... Kf8 2. Kd7), and now 3. e6 4. e7 and 5. e8=Q are ineveitable.
The only other question about this "all three rules apply" scenario might be to ask what happens if the pawn is not so far advanced. Say the kings are still on e6 and e8, but the white pawn is only on e5. Black still must sidestep, and white can still just grab the queening square. For example 1. ... Kd8 2. Kf7 Kd7 {With the queening square coverd, black's only try is to attempt to get closer to the temporarily undefended pawn, running away does nothing} 3. e6 {and the pawn is defended, making the rest trivial} Kd8 4. e7+ and 5. e8=Q
And if the pawn is further back still, say on e4, then white just brings it forward until the first scenario is reached. Say 1. ... Kf8 2. e5 Ke8 {trying to take back the opposition, but of course to no avail} 3. e6 and we're back to the beginning.
So what if only two conditions are met? Well there are three cases, so let's look at them one by one.
1) King is in front of pawn with opposition, but king is NOT on the 6th rank:
An example position might be a white pawn at e4, white king at e5, and black king at e7 - black to move. Just like before the black king cannot come forward, due to the opposition. In this case it's not yet on the eigth rank and so technically has two options - sidestep or retreat. Retreating, however, just brings us into "all three rules are met" and we know that loses (1. ... Ke8 2. Ke6 and see above). So side-stepping is the only try, but just like before, whichever direction the black king side-steps to, white can come forward to the other side.
Say 1. ... Kf7 2. Kd6 and now what can black try? White threatens Kd7 to claim the queening square and win. The only way black can prevent this at the moment is 2. ... Ke8 {briefly grabbing control of d7} 3. Ke6 and white wins because all three rules apply.
So the point is that if the king is in front of the pawn, and has opposition, he doesn't need to be on the sixth rank *yet* because he will inevitably get there, at which point all three rules will apply, and the pawn will queen.
2) King is in front of the pawn and on the 6th rank, but without the opposition.
An example position might be a white pawn at e5, white king at e6, and black king at e8, but this time it is WHITE to move.
Well now it is white who is forced to sidestep with his king, but it's a lot less harmful than it was for black. Consider this line: 1. Kd6 Kd8 {black's only hope here is that he holds the opposition, so presumably he'll try to retain it, the other "try" of 1. Kd6 Kf7 2. Kd7 gives up the queening square, and the pawn is close enough to the white king that it can't be picked off} 2. e6 and now white has his king on the 6th rank WITH the opposition (thanks to the tempo gained by pushing the pawn), but no longer has his king in front of the pawn. In other words, this will be case 3, and we'll return to it momentarily.
What if the pawn was farther back? Would that allow black's alternate "try" to have a chance? Well, no. For example a pawn on e4, king on e6, and black king on e8 just leads to 1. e5 gaining the opposition and giving us "all three rules" again.
3) King is on the sixth rank, with the opposition, but not in front of the pawn.
And therefore this is the only case left. Let us revisit the position we had reached in case 2 after white played 2. e6; a white pawn on e6, king on d6, and the black king on d8 - black to move.
Just like the other 6th rank with opposition scenarios, the black king must of course side-step, and Kc8 obviously just loses to any number of white responses. So black clearly MUST try to step back in front of the pawn. 1. ... Ke8 2. e7 {defended by the white king, and cutting off both d8 and f8 from black's king} Kf7 {only legal move} 3. Kd7 and white wins.
Remember though, all of these "three rules" situations rely on one thing: the pawn can't be a rook pawn!