Learn Finnish: Meidän piilomme on alakerrassa. - Our hiding place is downstairs.

Because it describes a static location (answering the question "where?"). Finnish spatial words follow a three-way pattern: being somewhere, going there, and leaving from there. Since the hiding place is simply sitting there, we use the location ending -ssa.

If we were running downstairs to the hideout, how would the word alakerrassa change?

If there were movement towards the destination, you would use the "where to?" form. It would become alakertaan. If you were coming from downstairs, you would use the "where from?" form, which is alakerrasta.

Why does the word alakerta change to alakerrassa with a double "r"?

This is consonant gradation. The word is a compound of ala- (lower) and kerta (floor). Adding the closed -ssa syllable at the end causes the strong rt combination in kerta to weaken into a double rr.

Do we really need both meidän and the -mme ending on piilomme?

In written and standard Finnish, yes. When you use the genitive pronoun meidän (our), the noun it describes must take the matching possessive suffix -mme. However, in casual spoken Finnish, people usually drop the suffix and just say meidän piilo.