Can I get a widebar without using the mathabx package?

1) You can use only widebar through:

% from mathabx.sty and mathabx.dcl
DeclareFontFamily{U}{mathx}{hyphencharfont45}
DeclareFontShape{U}{mathx}{m}{n}{
      <5> <6> <7> <8> <9> <10>
      <10.95> <12> <14.4> <17.28> <20.74> <24.88>
      mathx10
      }{}
DeclareSymbolFont{mathx}{U}{mathx}{m}{n}
DeclareFontSubstitution{U}{mathx}{m}{n}
DeclareMathAccent{widebar}{0}{mathx}{"73}

or simpler (with example):

documentclass{article}
DeclareFontFamily{U}{mathx}{hyphencharfont45}
DeclareFontShape{U}{mathx}{m}{n}{<-> mathx10}{}
DeclareSymbolFont{mathx}{U}{mathx}{m}{n}
DeclareMathAccent{widebar}{0}{mathx}{"73}
begin{document}
$widebar M = widebar{abcd}$
end{document}

2) I don't think it matters much, bar h is also higher than bar a. Just remember not to use bar M and widebar M together.

3) Maybe bad hinting. I've no idea.

I show my solution for this problem for plain TeX, where amsmath.sty isn't loaded, because the first line of this macro file says: NeedsTeXFormat{LaTeX2e}. I leaved the skewchar calculation because this is more suitable for accents like dot. My ẅidebar begins by left slanted border (by default) and ends at the same place as overline. If the first token is non-letter then normal overline is used.

But there are many exceptions for this rule declared in widebarE macro. If you feel that you are able to do better look of widebar, then you can simply re-define the widecharE or add to this macro more exceptions for more characters.

newcounttmpnum newdimentmpdim
{lccode`?=`p lccode`!=`t  lowercase{gdefignorept#1?!{#1}}}
edefwidecharS{expandafterignoreptthefontdimen1textfont1}

defwidebar#1{futureletnextwidebarA#1widebarA}
defwidebarA#1widebarA{%
   deftmp{0}ifcatnoexpandnext Adeftmp{1}fi
   widebarE
   ifdimtmp pt=0pt overline{#1}%
   else {mathpalettewidebarB{#1}}fi
}
defwidebarB#1#2{%
   setbox0=hbox{$#1overline{#2}$}%
   tmpdim=tmpht0 advancetmpdim by-.4pt
   tmpdim=widecharStmpdim
   kerntmpdimoverline{kern-tmpdim#2}%
}
defwidebarC#1#2 {ifx#1end else 
   ifx#1nextdeftmp{#2}widebarD 
   elseexpandafterexpandafterexpandafterwidebarC
   fifi
}
defwidebarD#1end. {fifi}
defwidebarE{widebarC A1.4 J1.2 L.6 O.8 T.5 U.7 V.3 W.1 Y.2 
   a.5 b.2 d1.1 h.5 i.5 k.5 l.3 m.4 n.4 o.6 p.4 r.5 t.4 v.7 w.7 x.8 y.8
   alpha1 beta1 gamma.6 delta.8 epsilon.8 varepsilon.8 zeta.6 eta.4
   theta.8 vartheta.8 iota.5 kappa.8 lambda.5 mu1 nu.5 xi.7 pi.6
   varpi.9 rho1 varrho1 sigma.7 varsigma.7 tau.6 upsilon.7 phi1
   varphi.6 chi.7 psi1 omega.5 cal1 end. }

deftest#1{$let.=#1 .M, .A, .g, .beta, .{cal A}^q, .{AB}^sigma, 
  .H^C, .{sin z}, .W_{!n}$}

testwidebar

testoverline

bye

First line shows the widechar, second line is normal overline.

I confronted a similar problem with a need for a pretty bar over various characters with mathbf, mathcal, mathfrak, etc. By "pretty," I mean that the bar seems very appropriate to the character shape and size, and this means adjusting the length and placement (left and right offsets) until it looks right. Further, the commands used must be compatible with MathJax in HTML after the TeX file is converted to XML for conversion to HTML on the Springer website.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%        Program for "manually" adjusting the bar over capital letters     %
%        or any other character to put a bar over. You specify the         %
%        character and two offset lengths. The program calculates the      %
%        the bar length such that the character with a bar has the same    % 
%        same length as the character without a bar.                       %
%                                                                          %
%        Specify the character and initial offsets below. Produce a pdf    %
%        and zoom in to see the barred character full screen. Return to    %
%        the TeX file and change the offsets until you are happy with      %
%        with the resulting appearance. Then zoom out with the pdf viewer  %
%        and copy the command for pasting wherever you want to save it.    %
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
documentclass[letterpaper]{article}
usepackage[vmargin=0.25in]{geometry}
usepackage[tbtags]{amsmath}
usepackage{amsfonts,amssymb}
usepackage{calc}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%   If you want to use a character from another font (e.g., Euler)         %
%   then add the needed package above.                                     %
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

newlength{charlength}
newlength{leftoffset}
newlength{rightoffset}
newlength{overlength}
newlength{backlength}                             %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
                                                    %      User settings          %
newcommand{testchar}{mathcal{S}}                 % Set the character you want  %
newcommand{printchar}{textbackslash mathcal{S}}% twice, slightly different.  %
setlength{leftoffset}{0.71pt}                     % Set the left offset and the %
setlength{rightoffset}{0.71pt}                    % right offset for the bar.   %
                                                    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
settowidth{charlength}{$testchar$}
setlength{overlength}{charlength - leftoffset - rightoffset}
setlength{backlength}{charlength - rightoffset}

pagestyle{empty}
begin{document}
vspace*{25mm}

$overset{kernleftoffsetunderline{kernoverlength}kernrightoffset}{testchar}$

$text{textbackslash overset{textbackslash kerntheleftoffsettextbackslash underline{textbackslash kerntheoverlength}textbackslash kerntherightoffset}{printchar}}$

bigskip

$kernleftoffsetoverline{vphantom{testchar}kernoverlength}kern-backlengthtestchar$

$text{textbackslash kerntheleftoffsettextbackslash overline{textbackslash vphantom{printchar}textbackslash kerntheoverlength}textbackslash kern--thebacklengthprintchar}$

end{document}

One drawback is that you must the character you want in the program twice in slightly different ways. For example, if you want a nice bar on mathfrak{C}, then in the program, which is now set for mathcal{S}, you must change S => C twice and change cal => frak twice.

But the payback for the double effort is that you get the command to save or put into another TeX file by copying the command from the pdf in your reader (I use SumatraPDF because the pdf file stays open in the reader while I change the TeX, and when I make the new pdf in WinEdt, it transfers me back to reader looking at the changed pdf in exactly the same place, etc., as when I left it to edit the TeX file).

The program gives two variants: with overset and with overline of a phantom. The separation from the character with overset is about 20% greater than the separation with overline. On my screen with the so-called widebar{mathcal{S}} occupying the full 23.5 cm height of the reader view, the separation is 3 cm for overset and 2.5 cm for overline.

Suppose you have adjusted the left and right offsets for a pretty barfrakC, but you now need a double bar. What to do?

Copy the line newcommandbarfrakC{...} and paste it in the TeX file, change the mathfrak{C} test character (and print character) to barfrakC, and so on to get what you need to paste into newcommanddbarfrakC{}.