Hey guys and @leandriis,
I could manage to solve my problems by myself.
Here is what the final code looks like....just in case you are interested:
It includes a way:
to solve the prob with white thin vertical lines between the color filled cells by using @leandriis code in this thread above
to only change the row height of the first row by using
\usepackage{stackengine} % Package to change row height
\newcommand\xrowht[2][0]{\addstackgap[.5\dimexpr#2\relax]{\vphantom{#1}}}
in the preamble and \xrowht{10pt}within the multicolumn curly brackets for text. (How to specify the row height in LaTeX table?)
- to add vertical space only between two specific cells with
\addlinespace[0.2cm] right after the command for a new tabular line (Inserting a small vertical space in a table)

\documentclass[a4paper,12pt, enabledeprecatedfontcommands, toc = sectionentrywithdots]{scrartcl} %Art des Dokuments, Schriftgröße etc. , idxtotoc, bibtotoc,
\usepackage[onehalfspacing]{setspace} %Zeilenabstand
\usepackage[hmargin={3cm,2cm},vmargin=2cm, includehead]{geometry}%Maße für die wissenschaftliche Arbeit
\usepackage[parfill]{parskip} %Bessere Seitenumbrüche
\usepackage[T1]{fontenc} %Silbentrennung
\usepackage[german]{babel} %Sprachanpassungen
\usepackage{ragged2e} %für Blocksatzschalter \justifying
% \usepackage[utf8]{inputenc} %Direkte Angabe von Umlauten möglich
\usepackage{csquotes}
\usepackage{graphicx} %Einfügen von Bildern
% \usepackage{tabularx} %Tabellen
% \usepackage{booktabs} %Weitere Möglichkeiten für Tabellen
\usepackage{here} %Lage von Tabellen und Bildern
\usepackage{physics}
\usepackage{amsmath}
\usepackage{mathrsfs}
\usepackage{textgreek}
\usepackage{textcomp}
\usepackage{amssymb}
% \usepackage[labelformat=simple]{subcaption}
\usepackage{float}
\usepackage{pdfpages}
\usepackage{lmodern}
\usepackage{enumitem}
\usepackage{parskip}
\usepackage[hyphens]{url} % be sure to specify the option 'hyphens'
% \usepackage[pdftex,bookmarksnumbered,hidelinks,breaklinks]{hyperref}
% \usepackage[superscript]{cite}
% ======SCHRIFTART=================================
% Schriftart Helvetica
\usepackage[scaled]{helvet}
\renewcommand\familydefault{\sfdefault}
% ===================================================
\usepackage[per-mode = reciprocal, output-decimal-marker={,}, exponent-product = {\cdot}]{siunitx} %Si-Einheiten
\usepackage[headsepline,automark, draft = false]{scrlayer-scrpage} %Für Kopf und Fußzeile und Seitenzahlen
\clearpairofpagestyles
\ihead{\headmark}
\rohead{\pagemark}
% \cfoot{\pagemark}
\addtokomafont{pageheadfoot}{\upshape}
\KOMAoptions{onpsinit={\linespread{1}\selectfont}}
%=======TABELLEN=====================================
\usepackage{booktabs, tabularx}
\newcolumntype{Y}{>{\centering\arraybackslash}X}
\usepackage{stackengine} % Für Zeilenabstand
\newcommand\xrowht[2][0]{\addstackgap[.5\dimexpr#2\relax]{\vphantom{#1}}}
\renewcommand\tabularxcolumn[1]{m{#1}}
\usepackage[version=4]{mhchem} % newest version
\usepackage{siunitx}% instead of units, which is deprecate
\sisetup{detect-weight,detect-mode}%Si-Einheiten
\usepackage[labelsep = period, skip=1ex, format=plain, font=small, labelfont=bf, justification = justified]{caption} % new
%\usepackage[labelformat=simple]{subcaption}
\usepackage{colortbl}
\definecolor{ExpBlau}{rgb}{0.11,0.19,0.46}
\definecolor{gr}{rgb}{0.90,0.90,0.90}
%=====================================================
%=====================================================
\begin{document}
\renewcommand{\tablename}{Tabelle}
\renewcommand{\figurename}{Abbildung}
\captionsetup[table]{justification = raggedright, font=normal, singlelinecheck=false}
\begin{table}[htb!]
\centering
\footnotesize
\setlength\belowcaptionskip{5 pt}
\caption{\textbf{Molekulare Eigenschaften der P(EO-\textit{co}-EPB)-Copolymere$^{\textcolor{ExpBlau}a}$}}
\noindent
\begin{tabularx}{\textwidth}{>{\centering\arraybackslash}m{.04\textwidth}>{\centering\arraybackslash}m{.185\textwidth}YYYY}
\multicolumn{1}{>{\centering\columncolor{gr}}m{.04\textwidth}} {\xrowht{10pt}Nr.}
& \multicolumn{1}{>{\centering\columncolor{gr}[\dimexpr\tabcolsep + 0.3pt\relax]}m{.185\textwidth}} {\xrowht{10pt}Copolymer}
& \multicolumn{1}{>{\centering\columncolor{gr}[\dimexpr\tabcolsep + 0.3pt\relax]}X}{\xrowht{10pt}$\textit{M}_\text{n, NMR}$ (\si{\gram\per\mole})}
& \multicolumn{1}{>{\centering\columncolor{gr}[\dimexpr\tabcolsep + 0.3pt\relax]}X}{\xrowht{10pt} mol\% EPB$^{\textcolor{ExpBlau}c}$} %
& \multicolumn{1}{>{\centering\columncolor{gr}[\dimexpr\tabcolsep + 0.3pt\relax]}X}{\xrowht{10pt}$\textit{M}_\text{n, GPC}$ (\si{\gram\per\mole})} %
& \multicolumn{1}{>{\centering\columncolor{gr}[\tabcolsep]}X}{\xrowht{10pt} \textit{Đ}}
\tabularnewline\addlinespace[0.2cm]
% SASSK
1 & ThatsALongWord & xxxx & x,x & xxxx & x,xx \tabularnewline % 113
2 & ThatsALongWord & xxxx & x,x & xxxx & x,xx \tabularnewline % 114
\hspace{ 0.5mm} 3$^{\textcolor{ExpBlau}b}$ & ThatsALongWord & xxxx & x,x & xxxx & x,xx \tabularnewline % 101
4 & ThatsALongWord & xxxx & x,x & xxxx & x,xx \tabularnewline % 111
5 & ThatsALongWord & xxxx & x,x & xxxx & x,xx \tabularnewline % 105
6 & ThatsALongWord & xxxx & x,x & xxxx & x,xx \tabularnewline % 112
\end{tabularx}
\vspace{0.6 \baselineskip} \
\justifying
$^{\textcolor{ExpBlau}a}$ Hello Hello Hello Hello Hello Hello Hello Hello Hello Hello Hello Hello Hello Hello Hello Hello Hello Hello Hello Hello Hello Hello Hello Hello Hello Hello Hello Hello Hello Hello Hello Hello Hello Hello Hello Hello Hello Hello Hello Hello Hello Hello Hello Hello Hello Hello Hello Hello Hello Hello Hello Hello Hello Hello Hello Hello Hello Hello Hello Hello Hello Hello Hello Hello Hello Hello Hello Hello Hello Hello Hello Hello Hello Hello Hello Hello Hello Hello Hello Hello Hello Hello Hello Hello Hello Hello Hello Hello Hello Hello Hello Hello Hello Hello Hello Hello Hello Hello Hello Hello Hello Hello Hello Hello Hello Hello Hello
\label{tab:Copo}
\end{table}
\end{document}
\noindentright before\begin{tabularx}. Also, it does not really make sense to use tabularx without anyXtype column. In the previous version of your question, there was anXtype column present. Your table currently is wider than the textwidth, since you did not take into account the value of\tabcosepwhen calculating the column widths. – leandriis Nov 26 '20 at 15:40\noindentfixed the gap between the table and the left page border but the overlap on the right side remains. I have updated the MWE and the output picture again. – Benson_G Nov 26 '20 at 15:43tabularxbecause I want to get rid of the white thin vertical lines between the color filled cells. – Benson_G Nov 26 '20 at 15:44tabularxis not the problem per se. The issue here is that you use it while not declaring anyXtype columns in your table. Replace all instances ofm{.3\textwidth}withXand your table should be just as wide as the textwidth. – leandriis Nov 26 '20 at 15:44tabularxis not the problem I would highly appreciated a solution using only `tabulator anything else which is simpler to use. :D – Benson_G Nov 26 '20 at 15:47m{.3\textwidth}byX? – Benson_G Nov 26 '20 at 15:53m{.3\textwidth}byXmakes it even worse but I think I most probably did it wrong. See updated MWE and output Picture – Benson_G Nov 26 '20 at 15:58