Great, so I will update ooffice.4ht with this change.

The paragraphs are added by the configuration, so you can just remove them, like this:

\Preamble{xhtml,ooffice}
\Configure{displayquote}{\EndP
\bgroup \Configure{HtmlPar}
{\EndP \HCode{<text:p text:style-name="quote">}}
{\EndP \HCode{<text:p text:style-name="quote">}}
{\HCode{</text:p>}}
{\HCode{</text:p>}}%
}{}{}{\egroup\noindent}
\begin{document}
\EndPreamble

This seems to remove all unnecessary paragraphs and disables the indent of the following paragraph.

Furthermore there is an empty paragraph after (and sometimes also before) the quotes. This should not be there IMHO, as the space is determined by paragraph separation.

Many thanks for the swift reaction. There is one glitch still: The source information (first optional argument) is on a paragraph of its own, but it should be appended to the quote.

I.e.

should render

displayquote displayquote displayquote displayquote displayquote displayquote
displayquote displayquote displayquote displayquote displayquote displayquote
displayquote. (Source)

rather than

displayquote displayquote displayquote displayquote displayquote displayquote
displayquote displayquote displayquote displayquote displayquote displayquote
displayquote.

(Source)

While the standard quote and quotation environments are exported to respective styles and formatted accordingly, csquotes' displayquote environment is exported to the text body style. Same applies to the other csquotes display environments (foreigndisplayquote, hyphendisplayquote, displaycquote, foreigndisplaycquote, hyphendisplaycquote).

This results in a rather tedious and error-prone task of finding and reformatting all quotations in ODT.

Could these environments be exported to dedicated styles (or simply quote style) in output, please?

See MWE.