Learn Russian: Эти крестьяне жили намного хуже.

Nouns describing people, like nationalities or social classes, that end in -анин or -янин are irregular. In the plural nominative, they drop the -ин and add -е instead of the usual -ы or -и. So крестьянин becomes крестьяне.

How would this noun look in the genitive plural, for example if I wanted to say 'a lot of peasants'?

For the genitive plural, words ending in -анин or -янин drop the -ин and take no ending at all (a zero ending). So 'a lot of peasants' would be много крестьян.

Why do we use намного here instead of just много to mean 'much'?

When you want to say 'much' before a comparative word like хуже (worse), you must use намного or гораздо. You cannot use много by itself to modify a comparative. Намного literally means 'by a lot'.

Is хуже an adjective or an adverb in this sentence?

It acts as an adverb here because it describes the verb жили (how they lived). The base adverb is плохо (badly). Luckily, irregular comparative forms like хуже (worse) and лучше (better) are used for both adjectives and adverbs, so you do not have to memorize separate forms.