will not charge; with red battery icon; plug phone in to power icon

I have a used first generation iPhone SE with 64gig of storage. a few weeks ago battery drained quickly after a charge. It then got difficult to charge. Would only charge with Apple charger. Now, it will not charge. Starts with discharged battery icon and plug into a power supply icon.

(1) Should the iphone run with a discharged battery and the phone plugged into power?

(2) Is this a sign that I need to replace the battery?

(3) Could it be a bad charging circuit?

Robert