Ok, here's a simple question - can an Action from a card be canceled by other Action or by a Tactic?

Can I trigger other Actions from a card that is in play to cancel other Action?

Yes to both. If the text of your card specifies that you can cancel an action, then you can cancel an action (i.e Bright Wizard apprentice)

I'm not aking about Bright Wizard Apprentice - his action is clear. I'm asking if I can respond to a opponets tactic or action with my action?

You can "respond" with another Action or Tactic, but there are few of them that can "cancel" another Action/card already on the stack, which is what you originally asked.