No the lug is not approved for that type of installation.
At to the grounding electrode conductors there needs to be a proper fitting installed where they exit the panel. There are no holes provided in any panel that I am aware of that were made for installing grounding electrode conductors. These are drain holes and are also drain holes in the meter base.
Anytime that a grounding electrode passes through a panel such as pictured it is required to bond to the panel at the point where it passes through although this is never or hardly ever enforced.
It would depend on where the panel was installed as to the protection of the NM cable.
If the NM cable in this picture is subject to physical damage, then so is the panel and it would also need to be protected. I cannot see the whole room, but from what I can see and what I know of laundry rooms, the NM cable in the picture is not subject to physical damage.
What about what looks to be a splice in the black conductor in that panel?