Question: The gemara Avodah Zara (20) explains that praising a non-Jew is a violation of "lo sechaneim", and this is quoted in the Rambam (Hil. Avodah Zara 10:6) and the Shulchan Aruch (Y"D 151, and Shach s.k. 18 says that this applies to all non-Jews).

However, it seems like we aren't so careful about this. After all, Chazal themselves praised Antoninus and Dama ben Nesinah, and it seems like no one, from Rishonim (such as the Rambam's praises of Aristotle) to today, has any serious issue with this. Does anyone know of a source that discusses these specific cases or has a list of 'kulos' that would cover the common practice?

I know of a few exceptions (such the Shulchan Aruch there allows it if it's meant as a praise of Hashem or Jews) and the most widely applicable is probably that of R. Moshe Shterbach (Teshuvos Vehanhagos 4:197) who allows praising a non-Jew whom one doesn't know personally, but it seems like there must be more leniencies.

Answer: 
The Rambam in Hilchot Avodat Kochavim 10:1 states that the seven nations are included in the prohibition of lo techaneim, but Jews and Gerei Toshav are not. This seemingly leaves generic idol worship up in the air. The context of the verse in Deuteronomy is clearly restricted to the seven nations, although most Rishonim seem to extend it to all Idolators.

Society today has few idolaters in the classical sense of the word, and with all the murkiness around the law, for the sake of peace (Eivah), like many other laws involving relationships with non-jews, many rabbanim I have come into contact with are lenient. (As per the Shach's klalei hora'ah that state in a time of need one can rely on even a da'at yachid (single opinion) in the Rishonim.
I knew a Rav in Jerusalem who would refer to Shakespeare and other great gentile authors, only semi-sarcastically, as "moreinu v'rabbenu." I've never heard that one isn't supposed to praise a gentile. We don't poskin directly from the gemara.