Learn Spanish (Latin America): El sindicato va a cancelar la manifestación. - The union is going to cancel the demonstration.

No, in Spanish, sindicato almost always refers to a labor union or trade union. If you want to talk about a criminal syndicate, you would use a phrase like crimen organizado or la mafia.

How would I refer back to "the union situation" using the grammar from this lesson?

You would say lo del sindicato. By using lo de (the matter of / the thing about) plus the noun, you can easily refer back to a known situation without explaining it all over again. Notice that de and el combine into del.

Does manifestación translate directly to "manifestation"?

It can, but in everyday Spanish, una manifestación is the standard word for a political protest, march, or public demonstration. It is exactly what English speakers mean when they say "a protest".

Why use va a cancelar instead of the future tense cancelará?

You can absolutely use the future tense (cancelará). However, in spoken Latin American Spanish, the "going to" structure (ir a + infinitive) is used heavily for planned actions or things happening in the near future, making it sound very natural here.