Learn Swedish: Ska ni behandla visumet snart? - Will you process the visa soon?

We use ska because processing the visa is an intentional action that the subject (ni) has direct control over. You would use kommer att for predictions or things outside of the subject's control, such as the weather.

Does behandla always mean to process a document?

No, behandla is a very versatile verb. While it means to process or handle when talking about applications and documents, it is also the standard verb for to treat (both in a medical sense, like treating a disease, and in a social sense, like treating a person well or badly).

Is visumet an en-word or an ett-word?

It is an ett-word: ett visum. You add -et to the end to make it definite: visumet (the visa). Unlike some Latin loanwords that drop the '-um' in the plural (like faktum becoming fakta), visum is a zero plural word, meaning it stays exactly the same in the plural: flera visum (several visas).

If I turn this into a statement starting with Snart (Soon), how does the word order change?

Because Swedish is a V2 (verb-second) language, the verb ska must stay in the second position no matter what starts the sentence. The statement would simply be: Snart ska ni behandla visumet.