Learn Russian: Сколько килограмм?

It actually has changed, but it's wearing a disguise! After сколько, we use the Genitive case. As you're learning, we use the Genitive singular for uncountable things (like сколько сока for 'how much juice'). But for countable things, we use the Genitive plural. For units of measurement like килограмм (and грамм), their Genitive plural form is often exactly the same as the basic dictionary form: килограмм. (You might also see the strict formal version, килограммов, but килограмм is extremely common).

Does сколько mean "how much" or "how many"?

It means both! Russian doesn't have separate words for "how much" and "how many." You use сколько whether you're asking about uncountable things (like "how much water?") or countable things (like "how many kilograms?").

Are there missing words in this Russian sentence? The English translation feels longer.

Yes! In everyday Russian, it's very common to drop words that are obvious from the context, including verbs (since Russian doesn't use "is" or "are" in the present tense anyway). If you're at a market buying apples, the seller will simply ask Сколько килограмм? instead of a full, wordy sentence like "How many kilograms do you want?".

How do I pronounce the double "м" at the end of килограмм?

You just pronounce it as a single, normal "m" sound. In Russian, double consonants at the very end of a word are not held longer or pronounced twice.