If you take a fight action, can you during that action, spend resources to bring in Skill, Asset or Event cards?

No. Playing an asset or event is an action by itself. You can however commit cards to a test for their icons, which is free.

Unless they are "fast" and it's the correct timing window. In that case, you could, couldn't you?

Unless they are "fast" and it's the correct timing window. In that case, you could, couldn't you?

Yes, if it was something like: "Fast. Play when you are taking a Fight action...". Or Fast effects that muck about with the Chaos Bag, like Lucky!.

Or really any applicable Fast card (or free triggered ability). You could, for instance, declare a Fight action, play Cryptic Research, then commit some of the newly-drawn cards to the test. You'd generally want to play Cryptic Research before declaring your action, of course, but the option's there if you need it for some reason.

A card with fast does not count as an action.

A card with fast does not count as an action.

Yes, but not all potential Fast actions can be played in the middle of a Fight action.

I wasnt the player of the beat cop but he had a fast action to do 1 damage would have cause an attack of opportunity?

The ability on the beat cop is a triggered ability. Any actions that doesn't cost an action will not trigger any attacks of opportunity.