Learn Swedish: Behöver vi något socker?

We use något because socker (sugar) is an "ett" word (ett socker). You use någon with "en" words, något with "ett" words, and några for plurals.

Does något mean "some" or "any" in Swedish?

It can mean both! In a question like this one, it translates naturally to "any". If you were making a regular statement, like Vi behöver något socker (We need some sugar), it translates to "some".

Why doesn't the Swedish sentence use a word for "do"?

Swedish doesn't use helper words like "do" or "does" to ask questions. Instead, you just put the main verb first. So, behöver vi literally translates to "need we", which is how Swedish asks "do we need".

Could I use lite (a little/some) instead of något here?

Yes, you can! Behöver vi lite socker? (Do we need a little sugar?) is very common and sounds natural. Using något just specifically emphasizes the "any" aspect: do we need any sugar at all?