Learn Finnish: Onko tämä asiakirja jo virallinen? - Is this document already official?

It is derived from the noun virka (an official post or office). When adding the adjective-forming suffix -llinen, the root goes into its weak grade (the 'k' disappears), turning the stem virka- into vira-, resulting in virallinen.

Given the current lesson, why isn't a word like "document" formed using the -nto / -ntö suffix?

The -nto / -ntö suffix creates abstract outcome nouns derived from verbs (for example, lausua "to state" becomes lausunto "a statement / official opinion"). Because a document is a concrete item rather than the abstract result of an action, Finnish expresses it with a compound of two existing nouns (asia + kirja) rather than a verb derivation.

Asiakirja literally translates to "matter book". How does that mean "document"?

Historically in Finnish, the word kirja didn't just mean a bound book; it was used for any formal written record or register. Combined with asia (matter, affair, or case), an asiakirja is simply a formal written record pertaining to a specific matter—a document.

Could I put jo at the very end of the sentence, like we often do with "already" in English?

You could, but it sounds slightly less natural. In Finnish, adverbs of time like jo are typically placed immediately before the word they modify or emphasize. Here, it highlights the transition to being virallinen (official), so placing it right before that adjective is the best word order.