You don't need to declare a subsection, raising it's counter by \setcounter{subsection}{1} or \stepcounter{subsection} would be sufficient.
But as the section counter resets the subsection counter, I would do it in the preamble together with removing it from the reset:
\usepackage{remreset}% tiny package containing just the \@removefromreset command
\makeatletter
\@removefromreset{subsection}{section}
\makeatother
\setcounter{subsection}{1}
This way that counter will stay at the value 1 and the circles are shown without any \subsection.
A minimal working example demonstrating the solution:
\documentclass{beamer}
\usetheme{Frankfurt}
\usepackage{remreset}
\makeatletter
\@removefromreset{subsection}{section}
\makeatother
\setcounter{subsection}{1}
\begin{document}
\section{Test}
\frame{one}
\frame{two}
\frame{three}
\end{document}
Output:
If you uncomment the command \setcounter{subsection}{1} and compile twice, you could see the original problem of missing circles.
beamer can't automatically wrap the mini frames into a new line if they don't fit into a single one. Instead, a new line is started for each subsection when using with the miniframes theme:
Unfortunately this doesn't work for the Darmstadt theme you are using. The reason for this is that Darmstadt loads the outer theme smoothbars, which is very similar to the miniframes theme, but installs smooth shaded transitions between the header and the content. However, it also uses beamer's compress option to get all the mini frames into a single line, so using multiple subsections doesn't work here. Removing this option is not a good idea, though, because the shading is hard-coded into the theme, so the header can't change its height automatically and will overflow if you use multiple lines for the mini frames:
So the only thing you can do is to replace the smoothbars outer theme with miniframes (producing the headline shown in the first image):
\documentclass{beamer}
\usetheme{Darmstadt}
\useoutertheme{miniframes}
\makeatletter
\beamer@compressfalse
\makeatother
\begin{document}
\section{Section 1}
\subsection{Subsection 1}
\frame{}\frame{}\frame{}
\subsection{Subsection 2}
\frame{}\frame{}\frame{}\frame{}
\subsection{Subsection 3}
\frame{}
\end{document}
Like this, you can wrap the mini frames into several lines by using multiple subsections, as shown in the example code.
Best Answer
The navigation symbols are those normally placed at the bottom. To remove the circles you mentioned you can use:
Maybe you could use my answer from here?