Colin,
can you add the lines or things you want to do to an attachment (some tex file or doc) - Tex filter is just sending the tags between delimiters (usually double dollars) to renderer (Mimetex or some local distribution of Latex) and you should have only latex or pure text inside delimiters.
If you have in editor for example html inside delimiters renderer tries to render it as text - not html. Latex does not understand html formatting.
And different distributions of (La)TeX use also different commands and symbols...
And if you want to show only syntax you can use backslash before dollar to render plain dollar $ as $$ \$ $$ (I have here spaces around \$ in ' \$ ' )
My local Vista is using MikTex ( and my plugindemo site is using TexLive ) so I think I can test any problems you have with MikTex but I just did not figure out the actual problem meaning what do you expect to happen with some tags - or latex images - or syntax ...
If you check file filter/tex/filter.php and functions string_file_picture_tex and tex_filter you notice that double dollars are not the only possible delimiters - in Wiki square parenthesis can cause some odd results etc.
More examples please - and for example a MS Word attachment where it might be possible for us to see what you hope to see...framed with red color 
) Ah - this Word document explains a lot.
When you take examples from tex files they usually have more commands for complete latex documents but tex filter is very simplyfied version of rendering Latex using some packages defined in settings of tex filter (administration menu).
If you want to use tex filter change single dollars to double dollars or use those square bracket delimiters (they don't work in wiki anyway).
But some tags like those equation tags don't look correctly aligned - not with MikTex or Tex Live - if they are used with Tex filter (first equation is left aligned and second goes to the right...). Still you could use other latex to produce same kind of content -
http://docs.moodle.org/en/Development:MediaWiki_TeX_test
has some examples that should work almost the same way in MikTex, Tex Live and TeTex - Mimetex is using many custom tags that other distributions can't use and on the other hand mimetex can't use all the packages, symbols and commands that other distributions can use. A little test:
HTML (italics):
If f(x) = 3x + 7 and g(x) = x + 4 then f(x) + g(x) = 4x + 11 and f(x)g(x) = 3x2 + 19x + 28.
Your one line example with double dollars and begin equation...end equation tags (note the first equation):
If $$ f(x)=3x+7 $$ and $$ g(x) = x+4 $$ then $$ f(x) + g(x) = 4x + 11 $$ and $$ f(x)g(x) = 3x^2 + 19x + 28 $$.
If $$ f(x)=3x+7 $$ and $$ g(x) = x+4 $$ then
\[ \begin{equation} f(x) + g(x) = 4x + 11 \end{equation} $$$
and $$$
\begin{equation} f(x)g(x) = 3x^2 + 19x + 28 \end{equation} \]
Centered example with begin array and end array:
And like you can see here font size is not very similar in theme as font of Latex tags with 120dpi density so you might try to use smaller density in settings of tex filter or bigger fonts (Times new roman etc) on pages where you use maths and change tex filter to render pngs instead of gifs (other posts on maths forum)
Writing maths has always been a pain in moodle but it is slow and painful even with Word equation editor ...
And note that you can't automatically use all packages of latex but you can add them to your tex filter settings. For example if you want to use colors you need to use LaTeX preamble like
\usepackage[latin1]{inputenc}
\usepackage{amsmath}
\usepackage{amsfonts}
\usepackage[usenames]{color}
\RequirePackage{amsmath,amssymb,latexsym}
to be able to render tags like
\displaystyle x_{1,2}=\frac{-b\pm\sqrt{\color{Red}b^2-4ac}}{2a}
or
\displaystyle \colorbox{red}{testing dollars}\sum_{i=1}^{p}\sum_{j=1}^{q}\sum_{k=1}^{r}\textcolor{blue}{a_{ij}} \frac{\textcolor{green}{b_{jk}}c_{kl}}{\colorbox{Orange}{ text}}
(double dollars or square bracket tags around them)
MikTex like Tex Live has a lot of such packages that people never use...