There isn't really a specific name for it. The border of a map extent is sometimes called a "neatline" although it isn't necessarily of the checked type.

The checked neatline is essentially an extension of the graticule, the coordinate grid, and it should line up with the graticule. As with the graticule itself it is a functional thing that says things about the map, as do compass roses, rhumb lines, scale bars, etc. It's entirely possible to say contradictory things about a map this way if you don't know what you are doing. It's kind of like not understanding how vehicles work, but knowing that you've seen vehicles that go under water, vehicles that have caterpillar treads, vehicles with turbojet engines, and vehicles with convertible soft tops. You could certainly draw something that has all of those properties, but if you understand the differences between a submarine, a tank, a jet airliner, and a sports car and why they have the traits that they do, then the combination very clearly doesn't make any sense.

Restraint is one of the most important tools for a cartographer. Knowing what NOT to put onto a map is in many ways more important than knowing what TO put on it.

That said, if your goal isn't so much to make a functional map, or even something that is a passable representation of one, so much as a picture that is an abstract evocation of "mappiness", then you are rather more free to toss things around willy nilly in the same way that if you were trying to create an abstract evocation of "vehicleness" rather than a convincing vehicle design, the convertible turbojet sports submarine tank might be a reasonable choice. I think actual maps are the best way to evoke "mapiness" but then I work win the geography field in real life.

As for the little circles at the intersections of rhumb lines, that doesn't make any sense to me. There really isn't anything significant about those points. Rhumb lines are just a reference for measuring bearings for marine navigation; the direction is all that's important.