"They have many dogs."
Translation:יש להם כלבים רבים.
It's not wrong. However, that is not how a Hebrew speaker would say it, and for two reasons :
First, the wotd combination we use to say that someone has something is יש ל. You don't reverse the order for no reason.
Second, it stresses the word להם, implying that while they have many dogs, someone else doesn't.
How do you figure either of them's an adverb?
The word המון has a unique noun meaning as in a crowd of people: קול המון כקול שדי. There isn't something equivalent for הרבה,
They're both used as adjectives, and adjectives in Hebrew also get used with omitted nouns, so אכלתי הרבה for "I ate a lot". But I don't know if if הרבה here is a noun, or if there's an implied noun (אוכל) and הרבה is only an adjective.
The difference can be seen pretty easily in Hebrew since adjectives come after a noun and adverbs before it, or after a verb. And of course, adjectives modify a noun and adverbs modify a verb.
In your example, הרבה is not a noun, nor does it stand instead of a noun. It is an adverb, because it follows a verb and is linked to it. I am not a Hebrew native speaker, but I do understand grammar, which follows the same patterns across languages.
אכלתי הרבה - clearly an adverb, because it modifies a verb אכלתי הרבה תפוחים - again an adverb, because it is before the noun and is linked with the verb אני אוכל תפוחים רבים - now this is an adjective - it goes after the noun you cannot say - אכלתי תפוחים הרבה
I have not yet encountered the word המון that much, so I can't really give any examples, but I would guess it would go something like this:
אכלתי המון פיצה - it doesn't go after the noun, it's linked to the verb so it is not an adjective ראיתי המון גדול - this is where it is a noun