Learn Italian: Manca lo scontrino di qualche prodotto. - The receipt for some products is missing.

The word qualche (some or a few) is a special adjective in Italian. It is invariable and must always be followed by a singular noun, even when it describes a plural amount!

Could I use alcuni here instead of qualche?

Yes, but you would have to change the noun to plural. Because alcuni behaves like a regular adjective, it must match the noun in gender and number. You would say alcuni prodotti. The phrases di qualche prodotto and di alcuni prodotti mean the exact same thing in Italian.

Why is the verb manca at the very beginning of the sentence?

In Italian, it is completely normal to put verbs like mancare (to be missing) or succedere (to happen) before the subject. The subject here is lo scontrino (the receipt). You could say Lo scontrino di qualche prodotto manca, but starting with Manca is the much more natural way to state this kind of fact.

A quick refresher: why is it lo scontrino and not il scontrino?

Masculine nouns that start with an s followed by another consonant (like the sc in scontrino) always take the definite article lo instead of il.