If a unit is focused to strike, receives the appropriate amount of focus tokens, and subsequently in the same engagement has his/her focus tokens removed by cards such as "Trust your instincts" or "Double Strike", does this allow that unit to focus to strike again in the same engagement? 

Would removal of focus tokens by either of those cards allow that unit to participate in a subsequent engagement against another objective in the same turn? (Not taking "Wookiee Navigator" into consideration).

I'm asking because I'm getting smashed in the face by the light side non-stop. We've been playing with a single core set (my second is coming in the mail) and my friend has used the Jedi preconstructed deck with a slight modification. He removed the objective set with Obi-wan, and subbed in the Ewok pod. I get locked down by ewoks he can continually regenerate with the objective and then pounded with Luke that he's able to remove focus tokens from willy-nilly with the above cards.

aphextheory said:

With either option, you can strike twice with the same character in a single engagement as long as they are triggered during the "resolve strikes" step of said engagement.

To answer your other question, only Trust your Feelings would work this way, since Double Strike is a reaction to a unit focusing to strike and cannot be used after the "resolve strikes" step, and, therefore, couldn't remove the focus token without forcing the unit to strike again in the same combat.

Concur with Stormwolf here. Anything that removes all force tokens from a character during the engagement resolution portion of the conflict phase must strike again, as you cannot complete that portion of the phase while there are still ready units. If you have any further questions, I might suggest that you look at the FAQ pg 7 regarding Red Two and the rulebook pg 23 and 31.