I ran into a trailblazer a few years ago, that had the ABS pump running. It was barely noticeable audibly. It too had a stuck brake caliper. Also, had drove the owner nuts, he had replaced all 4 calipers.
He had it towed to my shop. Upon initial viewing of the truck on my lift. I noted left and right calipers were reversed on front and rear, bleeder screws go atop the caliper not on the bottom. I had everything in working condition and sliding freely. Pressed the brakes without the truck running never could get it to stick. For some reason I cranked the truck on the lift and had somebody in the cab pressing the brakes for me. I noticed I felt a small vibration on the lines leading from the ABS, continued to feel around and found the pump was running. Pulled the fuse it quit put it back the brakes stuck, pulled it they worked. After several rounds of that, I decided the module must be the cause. I don't remember the complete diagnostics, as its been a few years. I do remember the pump was bad, and possibly the whole ABS module and hydraulic body. I DO REMEMBER THE brake fluid was BLACK with a rust tint. I blamed the rust as the culprit.
As far as the 2 rubber hose looking pieces (obo 1inch long) on each front caliper. They stick inside the caliper pin bore, cushioning the caliper pin. They should be pushed fully into the caliper pin bore. Then the rubber boot installed. Then the pin lubricated and installed on the vehicle.