Learn French: Mes clés sont quelque part dans la maison.

You would wrap the verb sont with ne and nulle part. The negative sentence becomes Mes clés ne sont nulle part dans la maison. Notice that nulle part replaces quelque part completely.

What exactly does the word part mean in quelque part?

On its own, une part usually means a share or a piece, like a slice of cake. But in fixed expressions like quelque part (somewhere), autre part (elsewhere), and nulle part (nowhere), it translates more closely to a place or location.

Could I use à la maison instead of dans la maison here?

Using dans la maison emphasizes the physical building, meaning the keys are physically inside the house somewhere. If you said à la maison, it translates more to the general concept of at home, which sounds a bit less natural when you are actively searching for a physical object inside the building.

Does quelque part ever change its spelling to match the plural keys?

No, quelque part is an adverbial phrase, which means it is invariable. It never changes to match the gender or number of the subject, so it stays exactly the same whether you are talking about one key or multiple keys.