Learn Finnish: Laatikossa on pähkinättömiä keksejä. - There are nut-free cookies in the box.

This is a typical Finnish existential sentence (a "there is/are" sentence). When a sentence starts with a location (Laatikossa) and introduces an indefinite subject at the end, the verb is always in the third-person singular (on). It never becomes plural, even if there are multiple cookies.

Why are both "pähkinättömiä" and "keksejä" in the partitive plural case?

In an existential sentence, if the newly introduced subject is a countable plural noun, it must be in the partitive case to indicate an unspecified amount (some cookies, rather than all the cookies). Adjectives must always match the noun they modify, so the adjective also takes the partitive plural.

How does the basic dictionary word "pähkinätön" (nut-free) become "pähkinättömiä"?

Words ending in the "without" suffix -ton or -tön undergo a stem change before grammatical endings are added. The end of the word changes to -ttoma- or -ttömä- (with a double T). To make it partitive plural, we add the plural marker -i- and the partitive ending -ä, which combines to form pähkinättömiä.

Could I flip the word order and say "Pähkinättömiä keksejä on laatikossa"?