Do Secret Actions count?

Yes. The game will see a Secret Action played face down as an Action.

Do character abilities still trigger if the requirements are met through cards played but negated by the opponent or otherwise? For example, Wave 1 Thundercracker’s bot mode ability --> Player 1 uses Supercharge, but player 2 reveals Infiltrate and negates Supercharge. Does Thundercracker's ability still trigger because an action was played? Meaning he would indeed get +1?

The Infiltrate will scrap the action you played and negate its effect, but Thundercracker’s ability still triggers because you played an action.