The first problem (no "henceforth cited as..." for items cited only one time) was already solved here.
The second problem (no items cited only one time listed in Abbreviations) was already treated here (might this question be considered a duplicate of it?).
EDIT: Based on those, my solution follows (as suggested by moewe, I've changed my previous one to make it more elegant):
\documentclass{article}
\usepackage[style=verbose,backend=biber,citecounter=true]{biblatex} % option citecounter=true added
\addbibresource{biblatex-examples.bib}
\addbibresource{blref.bib}
% Set threshold to 1
\newcommand{\SHthreshold}{1}
% Redefinition of shorthandintro
\savebibmacro{shorthandintro}
\renewbibmacro*{shorthandintro}{%
\ifnumgreater{\value{citecounter}}{\SHthreshold}%
{\restorebibmacro{shorthandintro}%
\usebibmacro{shorthandintro}}%
{}}
% Consider only items cited more than threshold-value times for Abbreviations list
\defbibcheck{shorthand}{%
\iffieldundef{shorthand}%
{\skipentry}%
{\ifnumgreater{\value{citecounter}}{\SHthreshold}%
{}%
{\skipentry}}}
\begin{document}
Foo.\autocite{kant:kpv}
Bar.\autocite{kant:ku}
Baz.\autocite{kant:kpv}
\printshorthands
\printbibliography
\end{document}
I leave also my previous solution for completeness:
\documentclass{article}
\usepackage[style=verbose,citecounter=true]{biblatex} % option citecounter=true added
\addbibresource{biblatex-examples.bib}
\addbibresource{blref.bib}
% Set threshold to 1
\newcommand{\SHthreshold}{1}
\renewbibmacro*{cite:full}{%
\usebibmacro{cite:full:citepages}%
\printtext[bibhypertarget]{%
\usedriver
{\DeclareNameAlias{sortname}{default}}
{\thefield{entrytype}}}%
% \usebibmacro{shorthandintro}}% DELETED
\ifnumgreater{\value{citecounter}}{\SHthreshold}{% NEW
\usebibmacro{shorthandintro}% NEW
}{% NEW
}% NEW
}% NEW
% Category with citecounter > 1
\DeclareBibliographyCategory{allowshorthand}
\AtEveryCitekey{%
\ifnumgreater{\value{citecounter}}{\SHthreshold}{%
\addtocategory{allowshorthand}{\thefield{entrykey}}%
}{%
}%
}
\begin{document}
Foo.\autocite{kant:kpv}
Bar.\autocite{kant:ku}
Baz.\autocite{kant:kpv}
\printshorthands[category=allowshorthand] % option category=allowshorthand added
\printbibliography
\end{document}