My dog is frequently twisting herself around

It's almost like she's trying to gnaw or scratch at her butt. Her name is lily and she is an 8.5 year old Jackrussel-pug. She has also been biting and licking her paws along with the other 4 dogs in the house. I have the whole thing on video but it won't let me send it, this is the best picture I have.

Hello!  It sounds as though your sweet girl may be paying a lot of attention to her anal glands.  It is thought that anal gland disease is connected with skin allergy - and compulsive foot-licking can be too, so the two things may be linked.   Allergies are generally chronic in nature ie recur again and again over the course of a lifetime, so even if the signs appear to be mild then they may well be worth paying attention to.  For feet, walking the pet through water eg a footbath or rinsing with a bucket before coming into the house, can sometimes wash away allergens that are clinging to the fur and thereby help to releave the syptoms.   Other causes of bottom licking and circling can include lumps and bumps around the anal glands, constipation and skeletal or other sorts of pain.   Whatever thecause, your vet will be able to find more out by examining the patient and planning treatment from there.