Bits "peeling off" schrader valve?

It looks to me as though the pump head, the part that attaches to the tube, is damaging the valve stem of the tube. Personally I wouldn't worry much about some minor scuffing and gouging, as shown in your second picture, but when pieces start flaking off, as in your first picture, then I'd replace the tube, because it looks as though the valve stem might fail.

If the valve stem fails then the result will be a "blowout", meaning that there will be a loud bang and then you won't have any pressure. If you're riding at the time then you can do damage to the tire (or tyre if you prefer). In the worst-case scenario, a blowout could result in a crash if you were cornering at the time.

I don't think I've seen damage like that before. I would suspect the pump head first, especially since you reported trouble with it. Maybe you aren't using the pump correctly; the lever on the pump head is meant to be flipped one way to grip more loosely when you're pushing the pump head on the valve stem, and then flipped the other way to grip tighter when the pump head is in place. If you are not using the lever at all, or using the lever incorrectly, that may be part of the problem. With most pumps these days the lever is parallel with the rim to put the pump head on, and perpendicular to the rim to lock the pump head in place.

If the pump head is the problem, most bicycle shops will be able to sell you just the replacement head, which is cheaper than an entire new pump.

The type of damage shown is often due to under inflation or not centering the valve stem in the hole. When under inflated the tube doesn't exert enough pressure on the tire bead. This allows the tire to spin on the rim. That in turn causes the tube to shift pulling it at an angle and chaffing the valve stem.