When serching does it cost an action? I assume yes, but It doesn't say the words "as an action" when you click on a search token.

Yes. It is an action.

It is not a quest specific action, and therefore, is not given the "as an action" text, which is used in the standard play for other reasons that are also quest specific.

Also, If its on the reference card, its gonna cost an action. (Opening Doors, Search Tokens, Resting) whatever it might be.

In the Delve at least half of the search tokens don't mention that you can draw a search card, but you have to succeed a test to not suffer more bad consequences. Is this intended? Do you really don't draw search cards for them?, because I can't see how you run out of search cards when you get only 1 search card per stage.

Only draw a search card if the App tells you.

There is still a benefit to examining a search token even if you don't get a search card - the total cards you draw from the item deck at the end of the stage is increased by one for each search token you examined.