First time touching a carb. Going to attempt a repair on the Holley 2bbl.

To test the accel pump, open the choke and look down the bore while pulling the throttle cable open fast. You should see squirts of gas. Of course, do this with the engine cold or you could burn your face.

The "hard to start cold" problem probably isn't the accelerator pump, but rather the choke and associated linkage. If the engine was cold in the 1st photo, the choke isn't set correct and will be too lean when cold. The plate should be shut firmly by the rod coming up from the exhaust (adjustment on the thermal spring in exhaust pocket).

Once the engine fires and generates manifold vacuum, the "choke pull-off" should pop the choke open slightly (~0.25", don't quote me check the manual), otherwise it will idle too rich and probably die. The pull-off is the small vacumm can in the upper left of your 1st photo. It is common for the rubber diaphram to fail. Test it with a vacuum pump or pull the vacuum tube, push in the rod, and cover the hole with your finger and see if the rod stays in. You can buy just the pull-off.