Learn Swedish: Vi tvättar ofta många kuddar.

The singular form is en kudde (a pillow). Nouns that are en-words ending in -e belong to Group 2. To make them plural, we drop the -e and add -ar, giving us kuddar.

Both tvättar and kuddar end in -ar. Are they both Group 2 plural nouns?

No, tvättar is a verb meaning "wash" or "are washing". In Swedish, many present tense verbs end in -ar (like pratar or älskar). It is just a coincidence that this verb ending looks exactly like the Group 2 plural noun ending in kuddar.

In English we say "often wash", so why does the Swedish say tvättar ofta (wash often)?

Swedish uses the "V2 rule", meaning the verb usually has to be in the second position in a normal sentence. Since Vi is in position 1, the verb tvättar must take position 2. The adverb ofta (often) has to come after it.

Can I use mycket instead of många for "many"?

Not here! You must use många (many) because pillows are countable things. You would only use mycket (a lot / much) for things you can't count individually, like water or time.