We have a box that has both XLR and 1/4 in inputs. It allows instruments and mics to be plugged in and extended into the mixer and amp located in the back of the room. What do you call that thing? An audio box extender?

Is XLR superior to 1/4? The previous setup, the 1/4 in cable from the guitars were put into a transformer and fed into the the XLR to the mixer. What's the advantage of doing transformer to XLR vs just doing a direction 1/4 in cable into the mixer?

Is it a cable drum like these by any chance?

The short answer I think is no, they are exactly the same. However if one type is balanced and the other isn't, then the balanced one wins obviously. But the connector shouldn't make a difference. Personally I find sending anything other than mic signals down XLR cables plain confusing so I'd stick with instrument jacks.

The transformer was a DI box, explanation here. Basically it converts the high-impedance signal from the guitar into a normal low-impedance audio signal. This solves various audio problems, notably with bass guitars I think, as well as not trashing recording equipment that's not built for guitar signals. A lot of DI boxes have extra features too, like unbalanced-balanced conversions and ground loop isolators and such.
Quote by Tombe
With pedals you can throw your guitar down for an impromptu pedal drone solo, whereas if you did that on a rack it would just look like you were programming your washing machine.