Creating my own Beamer Theme

The following example should in principal display everything you mentioned. I redefined some of your elements, but now the positioning is very flexible. So if you want the frametitle a little bit further up, just play around with its coordinates.

[caution notice: all the coordinates are just to roughly fit our design, you might want to finetune them]

%texsx-example.tex
\documentclass{beamer}
\usepackage[utf8]{inputenc}
\usepackage[T1]{fontenc}
\title{There Is No ...}
\date[ISPN ’80]{27th International Symposium of Prime Numbers}
\author[author1 \quad|\quad author2]{Euclid of Alexandria \texttt{[email protected]}}
\institute{My University}

\usetheme{texsx}

\begin{document}

\begin{frame}
\titlepage
\end{frame}


\begin{frame} 
\frametitle{There Is No Largest Prime Number} 
\framesubtitle{The proof uses \textit{reductio ad absurdum}.} 
\begin{theorem}
There is no largest prime number. \end{theorem} 
\begin{enumerate} 
\item<1-| alert@1> Suppose $p$ were the largest prime number. 
\item<2-> Let $q$ be the product of the first $p$ numbers. 
\item<3-> Then $q+1$ is not divisible by any of them. 
\item<1-> But $q + 1$ is greater than $1$, thus divisible by some prime
number not in the first $p$ numbers.
\end{enumerate}
\end{frame}

\begin{frame}{A longer title}
\begin{itemize}
\item one
\item two
\end{itemize}
\end{frame}

\end{document}

%beamercolorthemetexsx.sty
\mode<presentation>

% Settings
\setbeamercolor*{title page header}{fg=white}
\setbeamercolor*{frametitle}{fg=white}
\setbeamercolor*{framesubtitle}{fg=white}
\setbeamercolor*{headline}{fg=red}
\setbeamercolor*{author}{fg=white}
\setbeamercolor*{date}{fg=white}
\setbeamercolor*{title}{fg=red}
\setbeamercolor*{institute}{fg=red}
\setbeamercolor*{item}{fg=orange}
% Items
\setbeamertemplate{items}[square]
\setbeamertemplate{sections/subsections in toc}[square]
\mode
<all>

%beamerinnerthemetexsx.sty
\mode<presentation>

% Frame title
\defbeamertemplate*{frametitle}{texsx}[1][]{
    \setlength{\unitlength}{1mm}
    \begin{picture}(0,0)(0,0)
        %author
        \put(10,2){%
            \usebeamerfont{footline}
                \usebeamercolor[fg]{headline}%
                \ifx\insertinstitute\empty\else
                    \insertshortinstitute\quad|\quad
                \fi
                \ifx\insertauthor\empty\else
                    \insertshortauthor
                \fi
        }%
        % Blue background
        \put(-10,-12){\color{aublue}\rule{\paperwidth}{12mm}}
        % logo
        \put(-10,-12){\includegraphics[height=12mm]{pic}}
        % Title
        \put(10,-8){%
            \begin{minipage}[c][30mm][c]{80mm}
                \ifx\insertframetitle\empty\else
                \usebeamerfont{frametitle}\usebeamercolor[fg]{frametitle}\insertframetitle\par%
                \fi%
                \ifx\insertframesubtitle\empty\else
                \usebeamerfont{framesubtitle}\usebeamercolor[fg]{framesubtitle}\insertframesubtitle\par%
                \fi%
            \end{minipage}
        }
    \end{picture}
}

\mode<all>

%beamerthemetexsx.sty
\mode<presentation>

% Requirement
\RequirePackage{tikz}
\RequirePackage{xcolor}

%Colour
\definecolor{aublue}{HTML}{0A1439}

% Settings
\useinnertheme{texsx}
\useoutertheme{texsx}
\usecolortheme{texsx}
\usefonttheme{texsx}
\setbeamertemplate{navigation symbols}{}
\setbeamertemplate{blocks}[rounded][shadow=true]  

\mode<all>

%beamerouterthemetexsx.sty
\mode<presentation>

% Title page
\defbeamertemplate*{title page}{texsx}[1][]
{ 
        \setlength{\unitlength}{1mm}
        \begin{picture}(0,0)(0,0)
            % Blue background
            \put(-10,-46.5){\color{aublue}\rule{\paperwidth}{72mm}}
            % logo 
            \put(0,28){\includegraphics[width=.15\textwidth]{pic}}
            % Title
            \put(23,32){%
                \begin{minipage}[c][30mm][c]{80mm}
                    \usebeamerfont{title}\usebeamercolor[fg]{title}\inserttitle\par%
                \end{minipage}
            }
            % author etc
            \put(23,-5){%
                \begin{minipage}[c][30mm][c]{80mm}
                    \ifx\insertauthor\empty\else
                        \usebeamerfont{author}\usebeamercolor[fg]{author}\insertauthor%
                        \vskip0.2cm%
                    \fi%
                    \ifx\insertdate\empty\else
                        \usebeamerfont{author}\usebeamercolor[fg]{author}\insertdate%
                    \fi%
                \end{minipage}
            }       
            % author etc
            \put(23,-53){%
                \begin{minipage}[c][30mm][c]{80mm}
                \ifx\insertinstitute\empty\else
                    \usebeamerfont{institute}\usebeamercolor[fg]{institute}\insertinstitute%
                \fi%
                \end{minipage}
            }               
    \end{picture}
}

\mode
<all>

%beamerfontthemetexsx.sty
\mode<presentation>

\setbeamerfont{title}{series=\bfseries,size=\fontsize{18}{22}}
\setbeamerfont{footline}{size=\fontsize{6}{8}}

\mode
<all>

enter image description here


Here's another solution with similar result but with mostly other methods. Hopefully, both solutions are useful for anyone constructing a new theme.

Also, I used \begin{frame}[plain]\titlepage\end{frame} to get rid of the headline from the title page.

The result:

Screenshot of result

%beamerthemetexsx.sty
\mode<presentation>

% Requirement
\RequirePackage{tikz}
\RequirePackage{xcolor}

%Colour
\definecolor{aublue}{HTML}{0A1439}

%logo
\pgfdeclareimage[height=1.5cm]{Logo}{example-image-a}%Loads logo here

% Settings
\useinnertheme{texsx}
\useoutertheme{texsx}
\usecolortheme{texsx}

\setbeamertemplate{navigation symbols}{}
\setbeamertemplate{blocks}[rounded][shadow=true]  

\mode<all>

%beamercolorthemetexsx.sty
\mode<presentation>

% Settings
\setbeamercolor*{title page header}{fg=white}
\setbeamercolor*{author}{fg=white}
\setbeamercolor*{date}{fg=white}
\setbeamercolor*{item}{fg=orange}
%\setbeamercolor*{logo in frametitle}{bg=magenta}
\setbeamercolor*{titleframe}{bg=aublue,fg=white}
\setbeamercolor*{info in head}{bg=,fg=red}
\setbeamercolor*{title page header}{fg=cyan}
\setbeamercolor*{framesubtitle}{fg=white}
% Items
\setbeamertemplate{items}[square]
\setbeamertemplate{sections/subsections in toc}[square]
\mode
<all>

%beamerouterthemetexsx.sty
\mode<presentation>

\setbeamertemplate{background}{
  \begin{tikzpicture}
  \useasboundingbox (0,0) rectangle(\the\paperwidth,\the\paperheight);
  \fill[color=aublue] (0,0.75) rectangle (\the\paperwidth,8);
  \ifnum\thepage>1\relax%
   \fill[white,opacity=1] (0,0) rectangle(\the\paperwidth,\the\paperheight);
   \fi
  \end{tikzpicture}
}

% Title page
\defbeamertemplate*{title page}{texsx}[1][]
{\vskip5pt\leavevmode% 
\begin{beamercolorbox}[wd=0cm,sep=0pt,ht=1.5cm,#1]{title page header}%
\pgfuseimage{Logo}%Inserts logo here, the logo is loaded in beamerthemetexsx.sty (so that the logo can be used both in the inner and outer theme)
\end{beamercolorbox}%
\begin{beamercolorbox}[wd=12cm,leftskip=3cm,sep=0pt,ht=1.5cm,#1]{title page header}%
      \usebeamerfont{title}\inserttitle\par%
        \ifx\insertsubtitle\@empty\vskip1.5em%
        \else%
            \vskip0.25em%
            {\usebeamerfont{subtitle}\usebeamercolor[fg]{subtitle}\insertsubtitle\par}%
        \fi%     
    \end{beamercolorbox}%
    \vskip0.75cm%
    \begin{beamercolorbox}[wd=12cm,leftskip=3cm,#1]{author}
      \usebeamerfont{author}\insertauthor%
    \end{beamercolorbox}
     \vskip0.2cm%
    \begin{beamercolorbox}[wd=12cm,leftskip=3cm,#1]{date}
      \usebeamerfont{author}\insertdate%
    \end{beamercolorbox}
  \vskip6cm%
  \begin{beamercolorbox}[wd=12cm,leftskip=3cm,#1,center%center centers the content in the box (horizontally)
]{institute}
      \usebeamerfont{author}\strut\insertinstitute\strut%
    \end{beamercolorbox}
}

\mode
<all> 

%beamerinnerthemetexsx.sty
\mode<presentation>

\defbeamertemplate*{headline}{texsx}[1][]
{%\leavevmode%
\begin{beamercolorbox}[leftskip=.1\paperwidth,%
dp=1pt,left,#1]{info in head}%
\hspace{\Gm@lmargin}\strut|\quad\insertshortinstitute\quad|\quad\insertshortauthor\quad|\quad%
\end{beamercolorbox}%
% \vskip-1.95pt %if the background of "info in head" is set to something else than white, this line can be used to remove the whitespace 
}

\defbeamertemplate*{frametitle}{texsx}[1][]
{\leavevmode%
\begin{beamercolorbox}[wd=.1\paperwidth,%INPUT SUITABLE WIDTH
ht=1.5cm,dp=0cm,#1]{logo in frametitle}%
\vbox{}\hspace*{-\Gm@lmargin}\pgfuseimage{Logo}%\Gm@lmargin is the left text margin, the logo is loaded in beamerthemetexsx.sty
\end{beamercolorbox}%
\begin{beamercolorbox}[wd=.9\paperwidth,%INPUT SUITABLE WIDTH
ht=1.5cm,dp=0cm,sep=5pt,#1]{titleframe}%dp=depth,ht=height, wd=width.
\usebeamerfont{frametitle}\strut\insertframetitle\strut\par%
\ifx\insertframesubtitle\@empty%
      \else%
      {\usebeamerfont{framesubtitle}\usebeamercolor[fg]{framesubtitle}\strut\insertframesubtitle\strut}%
      \fi%
\end{beamercolorbox}%=
}

\mode<all>