1. I have two dance classes on Monday.

2. I have two dancing classes on Monday.
(which one is correct as a subject, dance or dancing?)

3 answers

dancing is an action verb not an adjective. Dance could be a noun or adjective depending what is used, so the first one is correct since its describing classes.
Anonymous is incorrect. When you use 'dancing,' the present participle of the verb, it can serve as an adjective.

In your sentences, 'dance' and 'dancing' are being used as adjectives to describe 'class,' not as subjects.

Both your sentences are correct, but the phrasing most commonly used is sentence 1.
Dance classes is the correct one