Yes, they should work together

This is (unofficially) clarified by Mark Seifter (one of Pathfinder's Designers) at paizo messageboards when discussing the effects of a Kineticist's Kinetic Blade with a conductive weapon. I will bold the part that is relevant to us.

Conductive has some really vague wording, and given the Kineticist has probably the most-complex at-will SLA, so this is definitely a pertinent question.

Here's a few things I've taken away from reading conductive like 5 times in a row to scour its wording. Note that this is a personal take, and by no means official:

1) I agree with your assessment here. Based on the exact way that you sort of trigger the Su or SLA after a successful hit, kinetic blade would not work because it is used specifically as part of certain actions to make the blade, rather than something that is delivered on a hit, so the same thing that helps it in action economy and AoOs harms it here.

2) I agree with you again. It seems to me that since an antipaladin should be able to choose cruelties when delivering his evil lay on hands with conductive, a kineticist should be able to apply non-form alterations to the conducted kinetic blast on the same grounds. Forms would be out because you've already hit at this point with the weapon, so that's the form the attack took.

3) Here's actually another thing (and a good lesson not to use passive if we can help it). "This weapon special ability can only be used once per round" is passive. It seems to be saying that a particular character can only use the conductive weapon special ability once per round, and that's the ruling that makes the most sense, but due to the passive here, it technically is possible to make the claim that it could mean varying things.

To explain why Kinetic Blade doesnt work, we have to understand how it is activated. You create a kinetic blade as part of your attack or full attack action, you don't have to hit the target to activate it, you simply wish for it to activate and it does, as long as you are attacking your target. You could miss the attack using the Kinetic Blade, for instance. The following text also disqualifies the conductive use for Kinetic Blade:

Even if a telekineticist uses this power on a magic weapon or another unusual object, the attack doesn’t use any of the magic weapon’s bonuses or effects and simply deals the telekineticist’s blast damage.

With that said, why does it work for Antipaladins? Because their Cruelties are applied on a successful touch attack. And that's all you need, the ability must be activated on a touch or ranged touch attack.

Touch of Corruption:

As a touch attack, an antipaladin can cause 1d6 points of damage for every two antipaladin levels he possesses.

Cruelty:

At 3rd level, and every three levels thereafter, an antipaladin can select one cruelty. Each cruelty adds an effect to the antipaladin’s touch of corruption ability. Whenever the antipaladin uses touch of corruption to deal damage to one target, the target also receives the additional effect from one of the cruelties possessed by the antipaladin. This choice is made when the touch is used.

Yes

As you intuit, since it is the same attack roll that you are simply rerolling, the effect of the third benefit of Sharpshooter would persist.