Learn Swedish: Är min ansökan godkänd? - Is my application approved?

It looks like it, but no! Ansökan is the indefinite form (an application). It belongs to a small group of nouns ending in -an (like en början, en tvekan). To say "the application," you just use ansökan as well. Also, since we have the possessive word min here, Swedish grammar requires the noun to be in the indefinite form anyway.

How would I make ansökan plural if I wanted to ask about multiple applications?

Nouns ending in -an have a special irregular plural form. They usually change the ending to -ingar. So "applications" becomes ansökningar. If you were asking about several applications, the sentence would change to: Är mina ansökningar godkända?

Why is the word for "approved" spelled godkänd and not godkänt?

Godkänd is a past participle, which is a verb form acting like an adjective. Just like a regular adjective, it has to agree with the noun it describes. Because ansökan is an en-word, we use the basic form godkänd. If you were asking about an ett-word, like ett lån (a loan), it would be: Är mitt lån godkänt?