Learn Swedish: Har du många syskon? - Do you have many siblings?

We use många because syskon (siblings) are things you can count (one sibling, two siblings). Use många ("many") for countable nouns, and save mycket ("much" or "a lot of") for uncountable things like water, coffee, or time.

Why doesn't syskon have a plural ending like -ar or -er here?

Syskon is actually plural in this sentence! It is an ett-word that ends in a consonant (ett syskon). In Swedish, these types of words stay exactly the same in the plural when you are just talking about them in general. So, one sibling is ett syskon, and many siblings is många syskon.

Where is the word for "do" in Har du...?

Swedish doesn't use "do" to make a question. To ask a yes/no question, you simply put the verb first. The statement Du har (You have) flips around to become the question Har du? (Have you? / Do you have?).

Is syskon used the exact same way as "siblings" in English?

Yes, but it is much more common! In English, it's normal to ask "Do you have any brothers or sisters?", and the word "siblings" can sound a little formal. In Swedish, syskon is the standard, everyday word everyone uses.