What is the role of 'a lot' and 'much' in my sentences?

Whether an adverb can modify a noun (pronoun) or not has been debated and there is no firm conclusion. For example,

Almost (or Nearly) everybody came.

Some argue that here almost or nearly are adjectives modifying the pronoun everybody. Some also argue that they are adverbs modifying everybody. But, others argue they are adverbs that modify the determiner every rather than the pronoun everybody. Just because everybody is a one word doesn't mean they can't modify every. English parts of speech are not very clear cut and there could be more arguments based on how you analyze the parts of speech. Your examples:

I earn a lot more money than my sister does.

I have a lot more things in common with my friends than my family.

I can't stand much more things of this.

I eat a lot less things / food than I used to.

As you can see, if you insert appropriate words after the determiner more and less, they are not pronouns any more. They are determiners (adjectives). Also, you need to note that more and less could function as adverbs. Therefore, it is very hard to say they modify (pro)nouns.

You shouldn't consider a lot and much as Adverbs. They're actually a different part of speech -- Quantifier -- which wasn't discovered until after Donatus specified the Eight Parts of Speech, around the Fourth Century. So you're not playing with a full deck.

Quantifiers are a type of Determiner, like articles; they indicate amounts and have very special syntax. Treating them like adverbs just makes more problems. Like the presuppositions underlying this question.

In comments BillJ wrote:

“A lot” is a noun phrase and “more” can be either a determinative, as here, or an adverb. In your examples “a lot more” is an NP in which the determinative “more” is functioning as a “fused determiner-head”, i.e. the functions of determiner and head are fused together into the single word “more” (cf. “a lot more money”). The same kind of analysis applies to “a lot less”.

It's crucial to grasp the difference between 'category' (noun, verb, adjective etc.) and function (subject, object, determiner, modifier etc.). In "I like him a lot", "a lot" is an NP (note that it has the article "the") functioning as an adjunct of degree. And in "I earn a lot more than ...", it's still an NP but this time its function is that of pre-head modifier to the fused determiner-head “more". There is nothing new in any of the terms I have used.