I am employed but way over my head in debt. Recently, my wife accepted a summons served by a police officer from a creditor I am unable to fully pay at this time.

What would my absence cause and what is the worst thing that could legally happen if I ignore the summons and don't show up on the scheduled court date?

Furthermore, can my wife accept a summons for me if she isn't the party a creditor is attempting to summon?

YOU NEED to go to the court date. IF you don't they can rule 100% in the collections favor.
If you go they may rule for a fraction of what the company wants.
I even know people that got off scotfree and the Judge threw the case out. I don't know how the people got away with it though.

The worst thing is to not go

The worst thing you can do is to ignore the summons. If you don't go the judge will rule for a default judgement and that is when the trouble will begin. Wage garnishments, if your state allows and also the judgement will allow
Your bank account to be frozen and your life will become a nightmare! As far as your wife accepting your summons I don't know about your state but in mine it can be left on the door or handed to anyone that answers the door.