SF + ARO + Guts Roll

I'd say you got the rules wrong :/

B declares Move, A declares Sup Fire ARO, B declares Fire: Sup Fire -3 applies IF B is inside the Range of Sup Fire from A. NOW, you calculate all mods for A and B, and THEN you make the rolls. IF A loses the Face To Face, and suffers an impact but does not go into a Null State (or IMM), and has no Courage (or skills that includes it), he gets to roll Courage (or chooses to fail voluntarily). Failing the Guts roll means losing Sup Fire, succeeding means A does not lose Sup Fire.

The mine is placed at any position of A's Movement (if A declared move). There is no way to actively prevent a troop from placing a Mine, repeater, etc... (passively you can, by having no enemy models but a Camo token there, forcing A to make an Intuitive attack to place the mine).

It is only Face to Face if A declared "smoke Dodge" (you should always declare that if you throw smoke anyway) and fulfills the Dodge Special Dodge requirements (B cannot have Multispectral Visor N2 or N3, the Smoke Template has to cut Line of Fire between A and all units he wants to dodge).
Please revise Turn Sequence, since you seem to have "Effects" confused (think of it as "consequences", not "modifiers").

If you can't improve your Cover you go Prone once you fail Guts, that is correct.
To go up or down without stairs or Climbing Plus you need an Entire Order, so no, failing Guts won't allow you to climb down. Jumping requires Super-Jump or an Entire Order, so no again.

 

1. Suppressive Fire state comes in during effects (step 8), Guts rolls are made during conclusion (step 9). So the enemy ARO would not suffer any penalty as Suppressive Fire state has not come into effect at Resolution (step 7) when rolls are made. Suppressive Fire State is cancelled when failing a Guts roll (FAQ), so the end result is unit A in cover but no Suppressive Fire state.
2. Unit A placed the mine (indicated where it will go) when declaring the "antipersonnel mine" skill, this would be done either at Declaration of the First Skill (step 3) or Declaration of the Second Skill (step 5) and the mine/camo marker is placed during Effects (step 8), before the guts roll is made or failed.
3. It is face to face in specific circumstances: i) the smoke grenade template must be placed so that it blocks LOF between A and B, ii) B must not have MSV2 or MSV3. There would be no MOD from the smoke as it doesn't come into effect until Effects (step 8), after you've made all rolls in Resolution (step 7). Do note, it is only face to face because of the special dodge rule that smoke ammunition has.
4. Not explicitly spelled out, but generally if you're told to move then you have to follow the general movement rules and as such cannot end in a position where your base is unsupported, so no you couldn't jump down. Climb would be up for a little debate, but I think not unless you have the climbing plus skill when I'm inclined to believe you could.

 

This isn't what he was asking. The active trooper is declaring SF (as in, "going into suppressive fire") and an enemy can see him so declares BS attack as an ARO. @Cartographer has it right. In general, don't do this.

2, 3, and 4 were also answered correctly (as far as I can tell) by Cartographer, but I just wanted to add that if you are on a 4" high pillar, then dropping prone will give you partial cover against anything on the ground that is not 4" tall.