Dimension Arrowhead Blocks

Are you using a custom block? It sounds like you are and it has elements on layers other than 0.

I have realized that I am hopelessly confused by the "By Layer, By Block, Layer 0" options. I am making a series of trial dwgs to try to understand the finer points.

yes, and remember the differences between ON / OFF and FROZEN / THAWED.

I have had difficulties that were similar.
create a dimension and explode it. then list the arrowhead or whatever symbol it is that is causing your confusion.
layer 0 has special properties. if memory serves, when your block is in layer 0 it is supposed to take on the properties of the layer it is inserted in.
I use _oblique for my block, and I think I had to put it in the same layer as the rest of my dimensions, DIM, for my own personal happiness.
in other words, it works for me. with the entity, the block insertion point for _oblique and the dimension all in my DIM layer, I short circuit those headaches and everthing is frozen or thawed when I think it should be. all together.
there was probably an easier way - the way it was supposed to be. :)

It's a little confusing at first, because they're all chameleons really. What color is a chameleon? It's not simple.
BYLAYER color and linetype, when in a block inserted on a layer "A", will *look* like the BYLAYER color and linetype of layer "A". It will look different inserted on layers that look different. Considerable flexibility.
BYBLOCK color and linetype, when in a block inserted on a layer "A", will *look* like whatever the current color and linetype setting are. (They can also be changes later whenever you like.) If the current color and linetype are both BYLAYER, it will look exactly like the BYLAYER example above, but if you change the block's color and linetype the BYBLOCK elements in it will change. You could make it look the same on any layer if you want, or even different from on another on the same layer. It is most flexible.
Assigning color, linetype, and layer(s)-other-than-0 will mean that the block will always look exactly the same, no matter where it is inserted, and you will only be able to control its visibility by freezing the insert layer, or the layer(s)-other-than-0. No flexibility.
Layer 0 is meant to be used for elements in blocks that you wish to behave exactly as if they were made on the layer on which the block is inserted, from the point of view of visibility. If a block is made of all layer 0 stuff, inserted on layer "A", and you turn off layer "A", the block will disappear. If the block elements were on layer"B", and you insert it on layer "A" (and layer "B" remains on and thawed) they would remain visible if you turn off layer "A" (because they are on "B", and the block insert layer is not being ignored). Freeze "A" and it will disappear because when you freeze the layer of an insertion, the whole block and al its contents is ignored. Turning a layer off only asks that elements on that layer not be displayed, and in the case where the block elements are on layer "B", only the block's insertion point is on "A".
I suggest you play around with the various permutations if it still isn't clear. Remember that FREEZE and OFF are not the same thing. OFF will unclutter you screen to some extent, depending on the way the drawing is structured, by not rendering elements on that layer. FREEZE will not recognize or render anything associated with a layer and speed up your computer performance.
In my business, most people draw using BYLAYER as a default, and therefore think that the main reason for layering is to make different-looking stuff. I use a combination of all the above strategies to get the most flexibility in my blocks and prettier plots, and use layering for visibility control.

I have fixed this problem now. I used to make my initial layout dwg with a lot of "working" dimensions on it which were later erased for the final version. I decided it would be better to put these dimensions on a different layer and then turn that layer off for the final version. The problem was that when I put the dimensions on a different layer and turned that layer off, the arrowheads were still visible. The problem was that my standard dimensions were created "by layer". I have now re-made my standard dimensions "by block" and it all works as desired. (Until the next time)

Your explanation of the behavior you describe cannot be. I don't think you've grasped what I wrote below.
BTW, why dimension twice?