Learn Swedish: Vilket stökigt rum han har!

It depends on the noun. The word for room is an ett-word (ett rum), so we must use the neuter form vilket. If it were an en-word like a car, we would use vilken (vilken bil!).

Why isn't there a word for "a" like in the English translation "What a messy room"?

In English, we say "What a...!", but Swedish doesn't use the article (en or ett) in these exclamations. You just put vilken, vilket, or vilka directly in front of the adjective and noun.

Why does the adjective stökigt end in a "t"?

Because it describes rum, which is an ett-word. In Swedish, adjectives get a -t added to them when they describe ett-words. The base form for this adjective is stökig (messy).

Since vilket is usually a question word, shouldn't the verb come before the subject (har han)?

If you were asking a question like "Which messy room does he have?", you would swap them and say har han. But here, vilket is used to make an exclamation. Since it is a statement expressing surprise rather than a real question, the subject (han) comes before the verb (har).