Can the impossible option be chosen? For example, if there are no equipment on the field and I use bitter strike, can i choose to destroy an ongoing card when there are no ongoing cards on the field?
Yes.
If something is required to be done for a secondary effect, it will outright say it. For example: "If you do..." or "If they take damage this way..."
Yes, you can do the 3 damage without destroying anything as long as there are no hero cards in play. The Sentinels rules say that if you cannot do the full effect of a card just do as much as you can.
If the wording had been "Destroy a hero ongoing or equipment. If you do then Mr. Fixer deals 1 target 3 damage." then you'd have to destroy to get the damage.
Note that if there is an equipment in play but no ongoings, you can't say "I'm gonna destroy an ongoing, oops there aren't any!" in order to get the damage for free (or vice versa). If there is a valid option in play, you have to choose it.
What Jimmy said. The other statements are not wrong, but that one seems to answer your question more precisely. When they say "destroy a hero ongoing or equipment card", it's not really a statement of "destroy a hero ongoing card or destroy an equipment card" (which I believe would let you avoid destroying equipment if you had no ongoings, and vice versa); it's more "destroy a card (which may either be a hero ongoing card or an equipment card)".
Just confirming that bitter strike makes bloody knuckles with a weapon like jack handle in play pretty awesome. DW Fixer can play bloody knuckles, use the jack handle to deal all non-heroe targets 5 damage (plus any outside boosts, eg. Legacy), and then destroy bloody knuckles to satisfy the second clause of bitter strike and not be susceptible to the extra two point of damage for the rest of the round. I want to fight Dawn with him, now, just to cut down citizens. :D
While the above is an ideal case, bloody knuckles becomes an easy, nearly cost-free, boost if I'm unstanding the rules correctly.
I did something like this but we were fighting against Voss. His huge army just kept coming back so I just waited around with my Jack Handle to knock them down while the other players worked on Voss.