I'm not too sure what you are getting at here but I will try and please note this is not an area of expertise for me. You seem to be referring to our observation of this expansion like you are seeing it as a human would observe it, and as if the edge of the universe emits some form of light that we observe. It is important that you realize that this RATE of expansion is a constant, imagine the second derivative of a function(in most cases).

I'm also not sure what you mean when you say " Then if we looked at an object halfway between us and the edge, then it would be expanding at a slower rate" why do you believe this to be the case.

Also, note that just because you are looking out into the universe you are looking back in time, personally I do not like the notion that time has a forwards and back that is quite a Newtonian idea, for example, if you were to leave earth and were in a different galaxy you could see the dinosaurs being exitict as the light has not travelled that far yet. This looking back in time seems to be quite different to how you refer to it.

I hope this gave you somewhat some more insight.