Qualified licenses are what Jefferson County issued years ago.
Many people just owned one gun, so the permit application asked for the serial number and description of those you were applying for permission to carry. IIRC, there was space enough to list three guns. This sufficed for many people, and your permit then allowed you to carry just the one (or ones) listed.
When I applied the first time (a really long time ago!), I asked them if they could put a longer list on my permit (I implied a LOT longer!). They told me not to worry; that they would simply make it an "Unrestricted" permit. This allows you to carry ANY legal gun concealed whether it was yours or not.
Now, the opposite seems to have been implemented in Jefferson County. The permit is now imprinted "Unrestricted." I am not sure if they can even provide one not marked like that.
I think that is what is meant by "Qualified." The qualifications set up by the sheriff could also be anything else he wanted to verify administratively, e.g. corrective lenses, etc. If he wanted (and it would have some justification) to restrict legal carry when you are using corrective lenses (like a driver's license does). But then, they would have to have some way to test and verify your need for said lenses.
The restriction on carry at a public demonstration came about after many armed confrontations with mobs facing down law enforcement. How many officers would stand up to an armed crowd that could outnumber them 10 to 1, or 100 to 1? This is codified in state law, and is not a qualification that is at the discretion of the local sheriff. Read the exact language you cite in your quotation and you'll see that the intent is demonstrations. I would add that the definition that includes sporting events is a stretch. The applicable code is in 13A-11-59. It specifically lists "Demonstrations" and gives further clarification. Here is the website for the complete code: