How to print only editor and title of parent entry in a child entry in the bibliography

To print the names you need \printnames{labelname}.

If you want labelname to print full names, then you need \DeclareNameAlias{labelname}{given-family}.

To get (ed./eds.) after labelname, add \setunit*{\addspace}\printtext[parens]{editorstrg}.

To get (ed./eds.) after the main editor in entry proc, you need to patch the bbx:editor macro using xpatch, like this:

\xpatchbibmacro{bbx:editor}
  {\usebibmacro{#1}}
  {\setunit{\addspace}\printtext[parens]{\usebibmacro{#1}}}
  {}{}

To get the the labeltitle in italics you can use \printfield[title]{labeltitle} since the default format for title is italics.

So your crossref:label macro becomes:

\newbibmacro{crossref:label}{%
  \entrydata{\strfield{crossref}}
     {\DeclareNameAlias{labelname}{given-family}%
      \printnames{labelname}%
      \setunit*{\addspace}%
      \printtext[parens]{\usebibmacro{editorstrg}}%
      \setunit*{\addcomma\space}%
      \printfield[title]{labeltitle}}}

Note: This is very specific to your question. More customisation is needed to handle cases when labelname does not refer to an editor (e.g., translator).

Here's the full MWE and output:

\documentclass[]{scrartcl}

\usepackage{xpatch}
\usepackage[backend=biber,style=authortitle-icomp]{biblatex}
\usepackage{filecontents}

\begin{filecontents}{my.bib}
@incollection{inproc1,
    Author = {Test Author},
    Crossref = {proc},
    Pages = {1--10},
    Title = {Testing the Title}}

@incollection{inproc2,
    Author = {Test Author},
    Crossref = {proc},
    Pages = {10--20},
    Title = {Testing the second Title}}

@collection{proc,
    Editor = {Senor Editor and Senora Editora},
    Publisher = {Any Publisher},
    Title = {My Proceedings},
    Year = {2013}}

@inproceedings{inproc3,
    Author  = {Nother Author},
    Publisher = {Nother Publisher},
    Title   = {In Some Other Proceedings},
    Maintitle = {Main Title of Other Proceedings},
    Year     = {2001},
}
\end{filecontents}

\xpatchbibmacro{bbx:editor}
  {\usebibmacro{#1}}
  {\setunit{\addspace}\printtext[parens]{\usebibmacro{#1}}}
  {}{}

\DeclareBibliographyDriver{incollection}{%
  \usebibmacro{bibindex}%
  \usebibmacro{begentry}%
  \usebibmacro{author/translator+others}%
  \setunit{\labelnamepunct}\newblock
  \usebibmacro{title}%
  \newunit
  \printlist{language}%
  \newunit\newblock
  \usebibmacro{byauthor}%
  \newunit\newblock
  \usebibmacro{in:}%
  \iffieldundef{crossref}
    {\usebibmacro{crossref:full}}
    {\usebibmacro{crossref:label}}
  \newunit\newblock
  \usebibmacro{chapter+pages}%
  \iffieldundef{crossref}
    {\usebibmacro{crossref:extrainfo}}
    {}
  \setunit{\bibpagerefpunct}\newblock
  \usebibmacro{pageref}%
  \usebibmacro{finentry}}

\newbibmacro{crossref:full}{%
    \usebibmacro{maintitle+booktitle}%
  \newunit\newblock
  \usebibmacro{event+venue+date}%
  \newunit\newblock
  \usebibmacro{byeditor+others}%
  \newunit\newblock
  \iffieldundef{maintitle}
    {\printfield{volume}%
     \printfield{part}}
    {}%
  \newunit
  \printfield{volumes}%
  \newunit\newblock
  \usebibmacro{series+number}%
  \newunit\newblock
  \printfield{note}%
  \newunit\newblock
  \printlist{organization}%
  \newunit
  \usebibmacro{publisher+location+date}}

\newbibmacro{crossref:label}{%
  \entrydata{\strfield{crossref}}
     {\DeclareNameAlias{labelname}{given-family}%
      \printnames{labelname}%
      \setunit*{\addspace}%
      \printtext[parens]{\usebibmacro{editorstrg}}%
      \setunit*{\addcomma\space}%
      \printfield[title]{labeltitle}}}

\newbibmacro{crossref:extrainfo}{%
  \newunit\newblock
  \iftoggle{bbx:isbn}
    {\printfield{isbn}}
    {}%
  \newunit\newblock
  \usebibmacro{doi+eprint+url}%
  \newunit\newblock
  \usebibmacro{addendum+pubstate}}

\addbibresource{my.bib}
\pagestyle{empty}
\begin{document}

Test \cite{inproc1} and \cite{inproc2} and \cite{inproc3}
\printbibliography
\end{document}