Learn Russian: У наших местных школ нет проблем.

They are both in the genitive plural because they describe школ, which is in the genitive case after the preposition у. The possessive pronoun наш takes the soft-sounding ending -их to become наших. The adjective местный has a hard stem, so it takes the hard ending -ых to become местных.

Why do the nouns школ and проблем look so short here? Where is the rest of the word?

Both words are in the genitive plural case. The preposition у triggers it for школ, and the negative word нет triggers it for проблем. Since their dictionary forms (школа and проблема) are feminine nouns ending in -а, they drop that final vowel entirely in the genitive plural, leaving what is called a zero ending.

How does this sentence mean "have no problems" when there is no verb for "have"?

Russian typically expresses possession using the "У + [owner] + есть + [object]" structure. However, when you want to say someone does NOT have something, the word есть is dropped and replaced by нет. Literally, the sentence is structured as "At our local schools there-is-no of-problems."

Is the letter in pronounced?