[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.