In order for the nodate string to also appear in the bibliography, redefine date+extrayear like this
\renewbibmacro*{date+extrayear}{%
\printtext[parens]{%
\iffieldsequal{year}{\thefield{datelabelsource}year}
{\printdateextralabel}%
{\printfield{labelyear}%
\printfield{extrayear}}}}%
This solution works with mergedate=compact (the default) in authoryear (and authoryear-like bib styles). For other mergedate settings, you will have to apply a different modification, the pattern, however, is always the same:
Get rid of the \iffieldundef{\thefield{datelabelsource}year} check and make sure that the second part of the code (the "else" statement) is executed (this can often be achieved by simply commenting out the \iffieldundef{\thefield{datelabelsource}year} line, since the "if" portion of the code is often an empty group and does nothing).
For the sorting, I would recommend
\DeclareSortingScheme{nyt}{
\sort{
\field{presort}
}
\sort[final]{
\field{sortkey}
}
\sort{
\field{sortname}
\field{author}
\field{editor}
\field{translator}
\field{sorttitle}
\field{title}
}
\sort{
\field{sortyear}
\field{year}
\literal{9999}
}
\sort{
\field{sorttitle}
\field{title}
}
\sort{
\field[padside=left,padwidth=4,padchar=0]{volume}
\literal{0000}
}
}
This is just the standard definition with \literal{9999} added to the end of the "year" sorting block.
So we make the year a 9999 for sorting, if it does not exist, of course this can be applied to other sorting schemes as well.
MWE
\documentclass{article}
\usepackage{csquotes,filecontents}
\usepackage[style=authoryear,dashed=false]{biblatex}
\begin{filecontents*}{\jobname.bib}
@BOOK{lennon,
AUTHOR = "John Lennon",
TITLE = "Peace on earth"}
@BOOK{lennon1972,
AUTHOR = "John Lennon",
TITLE = "More peace on earth",
YEAR = "1972"}
\end{filecontents*}
\addbibresource{\jobname.bib}
\DeclareSortingScheme{nyt}{
\sort{
\field{presort}
}
\sort[final]{
\field{sortkey}
}
\sort{
\field{sortname}
\field{author}
\field{editor}
\field{translator}
\field{sorttitle}
\field{title}
}
\sort{
\field{sortyear}
\field{year}
\literal{9999}
}
\sort{
\field{sorttitle}
\field{title}
}
\sort{
\field[padside=left,padwidth=4,padchar=0]{volume}
\literal{0000}
}
}
\renewbibmacro*{date+extrayear}{%
\printtext[parens]{%
\iffieldsequal{year}{\thefield{datelabelsource}year}
{\printdateextralabel}%
{\printfield{labelyear}%
\printfield{extrayear}}}}%
\begin{document}
\textcites{lennon}{lennon1972}.
\printbibliography
\end{document}
