So it looks like officer Wedge lets you spend squad tokens outside the normal time window (outside of activation). Since the card explicitly says you "spend" the token, does that mean you can trigger Rex (& other squad command effects) when you trigger Wedge if they're on the same ship?

No, thats not how it works.

Per RRG "Effects use and timing"

You are spending a squadron token to trigger Wedges card ability. It is being spent for that purpose and thus does not produce its normal effect.

Also, Rex requires you to resolve a squadron command for his ability.