Is Παιχνίδι the word you would use for a Play, as in dramatic production?
I've never come across it used in that context, but I'm not a native speaker, so I could be wrong.
The word I've encountered for plays is έργο.
You're right. Έργο (for television) or παράσταση (for theatre) is the word for a "dramatic production". ^.^
if you are talking about a play in the form of text, then your best choice with minimal context would be "θεατρικό έργο".