2. Does this "unknown substance" simply mean there was a spike in the chart that can't be attributed to any particular chemical they tested for, but that something was resident in Edelman's blood that isn't normal for humans to have?
Most likely, yes. A chromatogram such as below would have been generated. The people evaluating the data would try to identify what each peak corresponds to. Probably, a significant peak did not match any known substance.
3. And if so, could the unknown substance be something that's not illegal?
Yes, it could be. They also might suspect that it is a close family member of a known drug that is illegal, based upon where it appears in the spectrum above and based upon its molecular weight (the material is shunted to a mass spectrometer as each peak comes off the column.
4. And if so, wouldn't this circumstance already have been experienced and adjudicated by the NFL process, given that thousands of these tests have been done by now?
Yes, One would think that it would be extremely odd that Jules would have a substance in his blood or urine that had never shown up for any other player.