The "full" citation in the footnote is generated by the code
\usedriver{\DeclareNameAlias{sortname}{default}}{\thefield{entrytype}}
in \footfullcitetext. For a shorter citation this can be replaced by:
\printnames{labelname}\setunit{\addnbspace}\printfield[parens]{year}
Note that these short labels are open to ambiguity if a group of authors publish multiple works in the same year. To eliminate ambiguities, load biblatex with labelyear=true (or labeldate=true with version 2.6+) and use
\printtext[parens]{\printfield{labelyear}\printfield{extrayear}}
You've adopted the second alternative provided in the linked solution, where:
- footnote citations outside blocks must be added manually,
- recurrent citations must be tracked manually,
- regular footnotes aren't permitted.
So to address your second question, adapt the first alternative provided. Here is an example.
\documentclass{beamer}
\usepackage[style=numeric-comp,citetracker=true,sorting=none]{biblatex}
\usepackage{hyperref}
\usetheme{Berlin}
\setbeamertemplate{bibliography item}[text]
\renewcommand{\bibfootnotewrapper}[1]{#1}
\makeatletter
\let\cbx@citehook=\empty
\newtoggle{cbx@blockcite}
\renewcommand{\@makefntext}[1]{%
\noindent\normalfont\ifundef{\thefootnote}{}{\@thefnmark~}#1}
\DeclareCiteCommand{\sfcite}[\cbx@superscript]%
{\usebibmacro{cite:init}%
\let\multicitedelim=\supercitedelim
\iffieldundef{prenote}
{}
{\BibliographyWarning{Ignoring prenote argument}}%
\iffieldundef{postnote}
{}
{\BibliographyWarning{Ignoring postnote argument}}}
{\usebibmacro{citeindex}%
\ifciteseen
{\ifnumequal{\value{page}}{\csuse{cbx@page@\thefield{entrykey}}}
{}
{\ifnumequal{\value{framenumber}}{\csuse{cbx@frame@\thefield{entrykey}}}
{\usebibmacro{sfcite}}
{}}}
{\usebibmacro{sfcite}}%
\usebibmacro{cite:comp}}
{}
{\usebibmacro{cite:dump}}
\newbibmacro*{sfcite}{%
\csnumgdef{cbx@page@\thefield{entrykey}}{\value{page}}%
\csnumgdef{cbx@frame@\thefield{entrykey}}{\value{framenumber}}%
\xappto\cbx@citehook{\noexpand\cbx@footnotetext{\thefield{entrykey}}}}
\DeclareCiteCommand{\cbx@footnotetext}[\let\thefootnote\relax\mkbibfootnotetext]
{}
{\printfield[brackets]{labelnumber}%
\setunit{\addnbspace}%
\printnames{labelname}%
\setunit{\addnbspace}%
\printfield[parens]{year}}
{\multicitedelim}
{}
\newrobustcmd*{\cbx@superscript}[1]{%
\mkbibsuperscript{\mkbibbrackets{#1}}%
\iftoggle{cbx@blockcite}
{}
{\cbx@citehook%
\global\let\cbx@citehook=\empty}}
\def\cbx@beg@hook{\global\toggletrue{cbx@blockcite}}
\def\cbx@end@hook{%
\cbx@metabox{%
\global\togglefalse{cbx@blockcite}%
\cbx@citehook%
\global\let\cbx@citehook=\empty}}
\def\cbx@metabox#1{\edef\theprevdepth{\the\prevdepth}\nointerlineskip
\vbox to0pt{#1\vss}\prevdepth=\theprevdepth}
\BeforeBeginEnvironment{block}{\cbx@beg@hook}
\AfterEndEnvironment{block}{\cbx@end@hook}
\BeforeBeginEnvironment{beamerboxesrounded}{\cbx@beg@hook}
\AfterEndEnvironment{beamerboxesrounded}{\cbx@end@hook}
\makeatother
\addbibresource{biblatex-examples.bib}
\begin{document}
\begin{frame}{First Frame}
\begin{itemize}[<+->]
\item First citation.\sfcite{reese} Recurrent citation.\sfcite{reese}
\item Second citation.\sfcite{springer} Recurrent citation.\sfcite{springer}
\end{itemize}
\end{frame}
\begin{frame}{Second Frame}
\begin{beamerboxesrounded}[]{Beamer rounded box}
Recurrent citation.\sfcite{reese} New citation.\sfcite{companion}
\end{beamerboxesrounded}
New citation.\sfcite{bertram} Recurrent citation.\sfcite{bertram,companion}
Vanilla footnote.\footnote{Footnote text}
\end{frame}
\begin{frame}
\printbibliography
\end{frame}
\end{document}

Given that you're making the footnote citations "short", consider using the author-year style instead:
\usepackage[style=authoryear-comp,maxcitenames=1,uniquename=false]{biblatex}