This is NOT a Chihuahua. It is a Sphynx cat. A bald, grouchy Sphynx cat who will bite you if you mistake him for a Chihuahua.
I don't think there's a definitive answer in the rules or the FAQ, but I would certainly say "no" to both questions. I suppose you would not want to use a weapon (other than the flare gun) because you committed to a fight before your secret love lined up on the other side (or your secret hate lined up on your side), and you can better afford to lose a health point or two.
The reason I'd say "no" is because any item you carry can be displayed, just so that the Kid can't steal it using his special ability. Showing a weapon also serves as a warning to anyone who might want to pick a fight with you. But just because you've shown (say) a water canteen doesn't mean that you have to drink it next time you're thirsty. By the same logic, then, I would say that a weapon you've shown does not have to be applied to a fight.
I am curious to hear what Jeff has to say, though; he's pretty good about answering questions posted here.