Why are vertices not showing up when Blender says they are there?

[Brontobyte]

Both of these have vertices that I can not identify. The top one says it has 4 vertices and the bottom one says it has 36 vertices. How are vertices present but not shown in the viewport. ? When I select inverse I could delete them or I can dissolve by distance.
Question-2.82-4-19-2020.blend (615.7 KB)

On the top one I select 3 vertices and then I do a ctrl - I and delete the 4th vertex out but then one of the edges goes away and I can not figure out how to make the edge come back. Then I extrude a vert and suddenly it has 9 vertices. I am totally confused by this.
Why are there hidden vertices that I can not see. Question-2.82-4-19-2020.blend (615.7 KB)
Also I wanted to spin this object but mysterously I can not make it spin I wnat to make 8 copies but it just will not do it. ?
So 3 things I can not figure out.
Why are vertices present that I can not see. ?
Why wont this object spin ?
How to test if the mesh I made is legitimate. ?

[JohnMalcolm1970 (John Malcolm)]

From looking at the first blend file: Your triangle has 4 vertices. One of the is on top of another. Select All with Ctrl+A in Edit mode, then do Alt+M and By Distance to remove the double. The example you have beneath that is all sorts of screwed up. It has the same issue as the top triangle but also has a whole pile of verts on top of each other where the object’s origin is.

The second object is also acting incredibly weird because you have scaled it to a size of 0 in Z.

[Brontobyte]

CantjoinQuestion-2.82-4-19-2020.blend (643.8 KB)

So above I manually copied this and rotated them
now I want to join them
and this below is what happens

So for some reason this object has something wrong with it.

[Brontobyte]

Ok I see they need to have their transforms applied. This was created from a mesh that was from a P in edit mode. That Scale of zero must have been what gave it hell