[irach] No, not that kind of shoe. But shoes for dancing is still a valid general concept.
I am not sure if this will be helpful but I think I should add a note that the words on the card name something that may also be constructed from synthetic materials.
[INJ] Yes. Originally, the style of performance piece was strongly associated with one country. Then performers in a second country made a very strong contribution and a second strong association was created.
[Chalky] No, not a soft shoe shuffle.
[Raak] No, not reel shoes.

[Chalky] No, not ballet shoes.
[irach] No, not bluegrass clog dancing or bluegrass clog dancing shoes. But you are getting warm geographically. :-)

[Raak] No, not South American.

[irach] Yes, this particular style of footwear was far more likely to be worn by female dancers. I wouldn't swear to exclusively, though.

[irach] Yes, though not exclusively, the foremost association of the words on the card (and associated abstract concepts) is with the U.S.

[Raak] No, not square dancing.

[irach] No-no, not the can-can (or can-can footwear.)

[Software] No, not Flamenco dancing shoes. I think we covered that earlier.
[GL] No, not line dancing per se, though I believe there have been instances of this type of dance which involved lining up for a set of choreographed steps.
Here's a recap of what we presently know: the words on the card represent a type of footwear worn all but exclusively by women, while dancing in a particular way that is associated with another particular type of performance that is most closely associated with the United States but which also became associated with another country when performers from that country made a very strong contribution. Drums and tambourines drew favorable responses from the audience, which could suggest that what distinguishes the style of performance or dance is its rhythm or beat. You may also wish to consider that the kind of clothing worn on the foot is not always called a shoe. :-)

[irach] No, not related to belly-dancing.

[irach] No, not connected to salsa, mambo, cha cha, tango or any other Latin-identified style.
(I'm getting the feeling that the words on the card never made it too far outside of the U.S.)

[Raak] Yes. There is a strong African American influence/connection and I expect that influence traces back to Africa. *question engenders exuberant applause*

I've googled this, and that's all I can find.

[Raak] No, not clogs or taps.
*Wonders, should I offer a few suggestions to the players? Is that how this works?*

[irach] The wood is not a separate thing; it is what the heel of the footwear often is or appears to made from.

[irach] No, not jazz dancing shoes. The word "shoes" does not appear on the card at all.

[Raak] No, not pattens. I had to look that one up; like the words on the card, they are not so much in fashion these days. Come to think, it might help to keep in mind that dancing, music and fashion all change with the times.

Running out of synonyms for "clog"...