[bigd84]

My pipes go from 1/2'' to 3/4'', with the 3/4'' pipe capped at the top and measures an additional 12'' long. The lines going to the mixer valve come off the 3/4'' at the bottom, but are still 1/2''. Why is it plumbed in this manner? And should I plumb it the same when I change the mixer?

[KISS]

Yep, it's an air chamber. It helps to prevent water hammer and may or may not be required by code for toilets and showers particularly.

The chamber can be as long as you like. It's just a verticle column going nowhere but up. 3/4" gives you more air or cushion.

[speedball1]

KISS nailed it. This is a air chamber, Air chambers are there to cushion the shock of water hammer. In my area we are mandated to install air chambers on all lavatories and kitchen sinks. Over time they build up with condensate and must be recharged.
Hope this answers your question. Tom

[Milo Dolezal]

Agree with KISS and Tom. It is hammer arrestor. You can either purchase one or make your own.