Will a bladder infection cause you to hurt down your butt and leg?

Unlikely. Typical symptoms of a bladder infection are urinary frequency, urgency, burning, and bladder pressure and discomfort. Pain in the butt radiating down the leg unlikely and more suggestive of a sciatica-like problem. If truly a bladder infection with a positive culture, then an appropriate antibiotic will relieve all uti-related symptoms.
No. Pain in the distribution as you suggested describes the pattern of the nerve that travels in the leg called the sciatic nerve. The sciatic nerve is not the cause of the pain source, it is the result of an irritated nerve typically in the lumbar spine (low back) which are caused by herniated disks, spinal stenosis or degenerative disc disease requiring further evaluation by a spine specialist.