OK, let's see what we can do...
First off, re fallback, please provide details of what you actually did so that I can determine what file you are using (or you can send me a message with your URL
and I can poke at it) Fallback relies on a variable setting in asciimathml.js that is described in the docs. Frankly I don't recall what it is set to by default, but the first thing would be to double check and make sure that the fallback is indeed working. Once fallback is up and running you should be able to manage without the banner and without the plugin and without tex.
As far as tex is concerned you need to look at the docs: http://docs.moodle.org/en/TeX_filter#Debugging_TeX_filter
Your problem will be compounded by the fact that you are using WIndows as a server (I am allergic, don't you know) though Mauno runs a windows server and has played with the windows perms. But first thing is to run the texdebug and report the results. if the binaries are not there then you probably want to install texLive. If they are there but there are perm options then you need to set thenm correctly, and I think there is info on that in the forums (which should likely get included in the tex filter docs if it is not already....)
Now, you need not use tex and asciimath. Asciimath provides quite a few extra benefits over the moodle tex filter for many uers, especially if you are not pushing the limits of what tex can otherwise do, including asciisvg, calculators, chatmath, etc and it is extensible and works without a server. Asciimathml also provides a simpler syntax, which is why Peter says he started fiddling with it in the first place
As I mentioned, you can set dragmath to do single or double dollars and asciimathml will do latex based upon its inclusion in v2 (the version posted) of much of latexmathml.
Frankly, I would resolve the asciimathml fallback issues first and see if that satisfies you, and if so, turn off tex filter so that does not slow down display of pages, and if you are happy, set dragmath to a default output that you want.