Is there a connection between having a bad back with sciatica and cold feet?

Often times, Yes: Lower back pathology can cause pressure on nerve which innervates lower extremity and this can cause these symptoms.

Evaluation rec'd: Back pain causing leg pain and vascular problems causing leg pain can be very similar in presentation. It's never a bad idea to have a specialist physician differentiate the cause. It is important to make the distinction between your feet actually being cold vs "feeling " cold. Doctor evaluation warranted here if think.

Possibly: However there can be host of other reasons to have cold feet. I would suggest a pain/spine specialist that can help determine what is from your spine and what is not.