Min

What is meant by "active card"? The term does not appear in the rules.

Please refer to "Active effects" on Page 17. And an example on Page 7 (in the "resolve your card" step).

If Pyro, an enemy hero, and Min's mine are all in a straight line, can Widget's alt blue "Drag Off" push the enemy through the mine, causing it to go off and thus allowing Pyro to land where the mine was?

No. Movement is not done in separate steps. It's a single uninterruptable step from origin to destination (and since tokens are obstacles, you cannot end movement on a mine even if you can go through it to trigger it). You only check for mines exploding after the entire movement is done.

Min's push red says: after the attack, move one space to a space adjacent to the target, then push X spaces. If you target a minion and the minion dies, can you move to a space adjacent to where that minion was and push, or can you not move adjacent to the target since the target is now dead/gone?

No. After the attack there's no target if it was a minion (unless Brogan's or Tali's green is in effect).

Can I attack a minion with Min's primary red and then push an enemy hero who is next to Min?

Nothing is stopping you

If Min's initiative is reduced to a negative value, but he has activated the double item bonus of his green card, how is the outcome calculated?

Using the rules of arithmetic. Multiplication has higher priority than subtraction

If NebKher uses his gold card to perform Min's blue Mine card, does he just place illusion tokens instead of mine tokens, ignoring the rest of the text (which only describes the ability of the mine)? Or will the mine tokens also affect Min's teammates?

It would work as your friend suggests if Min's mines were based on an active effect, but they are not. Mine's effect is part of that token's _property_ and does not change once the token is placed.