Problem with lmodern font and paired delimiter / overline

Question

I'm currently writing my bachelor thesis and forgot to include \usepackage[T1]{fontenc}. But I'm using \usepackage[utf8]{inputenc} and \usepackage{microtype}. When I included \usepackage[T1]{fontenc} I needed to choose a scalable font, so I chose \usepackage{lmodern}. Now I have a few new problems. The hat from $\widehat{K}$is to large. I solved this with a new command:

\usepackage{accents}
\DeclareMathSymbol{\widehatsym}{\mathord}{largesymbols}{"62}
\newcommand\lowerwidehatsym{%
  \text{\smash{\raisebox{-1.2ex}{%
    $\widehatsym$}}}}
\newcommand\fixwidehat[1]{%
  \mathchoice
    {\accentset{\displaystyle\lowerwidehatsym}{#1}}
    {\accentset{\textstyle\lowerwidehatsym}{#1}}
    {\accentset{\scriptstyle\lowerwidehatsym}{#1}}
    {\accentset{\scriptscriptstyle\lowerwidehatsym}{#1}}
}

Now I have the following questions/problems:

1. I use $\fixwidehat{}$ with $\overline{\fixwidehat{K}}$ but the hat crosses the line.
2. My absolute-value-command \newcommand{\abs}[1]{\left\vert#1\right\vert} doesn't scale the size when $\fixwidehat{}$is used.
3. But $\abs{e^{i*t*\omega}}$ is compared to the normal computer modern font not well sized. It's to high.

I was very happy with my results before including \usepackage{lmodern} and just want to adjust it to look similar to the old version. Could you please help me? I have to print the thesis in a little more then a week and I'm currently really frustrated with LaTeX.

Edit: Here is a minimal working example:

\documentclass[
11pt,
a4paper,
twoside = false,
headings=standardclasses,
toc=chapterentrywithdots,
toc=bibliography]{scrbook}

\usepackage[T1]{fontenc}
\usepackage[utf8]{inputenc}
\usepackage[ngerman]{babel}
\usepackage{lmodern}

\usepackage{amsmath}
\usepackage{amsthm, thmtools}
\usepackage{amsfonts}
\usepackage{amssymb}

\usepackage{accents}
\DeclareMathSymbol{\widehatsym}{\mathord}{largesymbols}{"62}
\newcommand\lowerwidehatsym{%
  \text{\smash{\raisebox{-1.2ex}{%
    $\widehatsym$}}}}
\newcommand\fixwidehat[1]{%
  \mathchoice
    {\accentset{\displaystyle\lowerwidehatsym}{#1}}
    {\accentset{\textstyle\lowerwidehatsym}{#1}}
    {\accentset{\scriptstyle\lowerwidehatsym}{#1}}
    {\accentset{\scriptscriptstyle\lowerwidehatsym}{#1}}
}

\newcommand{\norm}[1]{\left\lVert#1\right\rVert}
\newcommand{\abs}[1]{\left\vert#1\right\vert}

\usepackage[pdftex]{graphicx}
\usepackage{tikz}
\usepackage[final,babel,tracking=true,kerning=true,spacing=true]{microtype}

\begin{document}
\begin{align*}
\overline{\fixwidehat{K}} \\
\abs{\fixwidehat{K}} \\
\norm{\fixwidehat{K}} \\
\abs{e^{it\omega}}
\end{align*}
\end{document}

Answer 1

You've been bitten by the problem described in Problems using lmodern with very small font sizes and the solution is similar.

\documentclass[
  11pt,
  a4paper,
  twoside = false,
  headings=standardclasses,
  toc=chapterentrywithdots,
  toc=bibliography,
]{scrbook}

\usepackage[T1]{fontenc}
\usepackage[utf8]{inputenc}
\usepackage[ngerman]{babel}
\usepackage{lmodern}

\usepackage{amsmath}
\usepackage{amsthm, thmtools}
\usepackage{amsfonts}
\usepackage{amssymb}

\newcommand{\norm}[1]{\left\lVert#1\right\rVert}
\newcommand{\abs}[1]{\left\vert#1\right\vert}

\usepackage{graphicx}
\usepackage{tikz}
\usepackage[final,babel,tracking=true,kerning=true,spacing=true]{microtype}

\DeclareFontShape{OMX}{cmex}{m}{n}{
  <-7.5> cmex7
  <7.5-8.5> cmex8
  <8.5-9.5> cmex9
  <9.5-> cmex10
}{}
\SetSymbolFont{largesymbols}{normal}{OMX}{cmex}{m}{n}
\SetSymbolFont{largesymbols}{bold}  {OMX}{cmex}{m}{n}


\begin{document}
\begin{gather*}
\overline{\widehat{K}} \\
\abs{\widehat{K}} \\
\norm{\widehat{K}} \\
\abs{e^{it\omega}} \\
X_{\widehat{x}_{\widehat{x}}}
\end{gather*}
\end{document}

For comparison, the output without the fix.

---

In my opinion you should not use automatic \left and \right for \abs and \norm: the second and third line show why. Check the documentation of mathtools for better methods. For instance, loading mathtools and issuing

\DeclarePairedDelimiter{\norm}{\lVert}{\rVert}
\DeclarePairedDelimiter{\abs}{\lvert}{\rvert}

you can look at the output of

\[
\norm{\widehat{K}}+\norm[\big]{\widehat{K}}
+\norm[\Big]{\widehat{K}}+\norm*{\widehat{K}}
\]

As you see, the first is quite right, maybe you can prefer the second, but the third and fourth (which corresponds exactly to your definition) have definitely too large fences.

Final note: I removed the pdftex option to graphicx; do it yourself as well. It's not related to the problem, but it's better policy.