I'll just give a quick response off the top of my head, and then maybe elaborate on it later when I have some more time...and have my new computer set up so I can actually run it. Besides, there seems to be several questions at the same time, so I might be addressing the wrong ones.
To count characters instead of words, you may use the option
-letter when running TeXcount. Note that, by default, spaces and punctuation are not counted. You can include punctuation amongst the counted characters by using the option
Digit,alphabetic is the default option. However, there's no way for TeXcount to actually count spaces, although you can get a rough estimate of the number of spaces from the number of words, assuming there is one space per word.
TeXcount still counts the number of equations, not the number of characters in the equations: perhaps not what you want. There's nothing that can easily be done about that in TeXcount.
As for grouping the counts into categories, TeXcount has some predefined rules: words/characters in main text, in titles, etc. as well as number of equations, floats. If these rules do not give to desired grouping, you'll need to define new macro handling rules for macros/environments you'd like to count differently; you can also define new counters in addition to the predefined ones. However, for now, I'll let that issue be.
What seems closest to the three groups in your example is words/letters in text, number of equations (which is divided into inline and displayed equations), and other words/letters which include footnotes but also captions. The names TeXcount use for these four are
As for templates for formatting the output, you might want to take a peek at a similar question: Using Texcount Templates
The simplest template could be to run TeXcount with options
which will write an output like
where the sum has been defined as
text+other+eq+ds (the four counters that have been given coefficients 1 in the
I'm not sure, but you may have to use the option
-tex since the template contains special TeX characters.
It might be more convenient to define the template in the document using the
%TC:template instructions (which TeX ignores as comments but TeXcount parses), or using a template file in which you can also specify the
When I get my TeX/TeXcount environment set up again, I might be able to help you out better: as of now, I'm not able to actually make a test example.