Thereupon and therewith are adverbs, right? I've read that it's okay to write "The class therewith gave the speaker a round of applause," so can I also write "The class thereupon gave the speaker a round of applause"? Are there adverbs you don't/don't usually write before verbs?
I wouldn't put them there. As 'thereupon' refers back to something mentioned, I'd put it at the beginning with a comma after it:
Thereupon, the class gave the speaker a round of applause
If you put it in the position you used, try putting commas around it.
Yes, adverbs often don't some before verbs. Sentence adverbs, which can be used to show the speakers opinion, are often placed at the start and separated from the rest of the sentence by a comma:
Surprisingly, they didn't do very well.
Also, in this 'very well' has an adverb that comes after the verb (well) and it is modified by another adverb (very). Adverbs have many different functions- they can modify adjectives, other adverbs and whole sentences as well as verbs.