It seems clear from the Rama, MB, and other sources (as well as from most printed siddurim) that Pirkei Avos is said in the synagogue at Shabbos Mincha as part of the tefillah in the summer months, and Barchi Nafshi in the winter. Are there any synagogues (other than yekke shuls) that still follow this? It seems from most siddurim that it is done before Aleinu (I also have seen this done at Breuer's, which is generally a good source for minhagim) so I'm specifically interested if there are other shuls that still do it this way?
The Great Synagogue of Rome still says Pirkei Avot between Mincha and Maariv as a communal liturgical recitation in the summer. Source: I was there once.
In light of edits to question: They did not do it before Aleinu IIRC, but they did do it before they broke for Seudah Shelishit, so I'd say it was still part of the general Mincha service and not part of the Maariv service.
Regarding Pirkei Avos:
Many (including Chabad) continue this chapter-a-week regimen throughout the summer months, until Rosh Hashanah.