You've got a point - in theory it would work based on the Finnish phrase but it isn't good English, then. One way to use the indefinite article would be then "A dirty and smelly dog." but that would also change the Finnish word order: "Likainen ja haiseva koira."
I see the articles still are not that self-evident for me :) I normally can hear already when it sounds correct and when not but every time I start to think in Finnish, I kinda lose the ability to think about the English articles, haha.
In Finnish it sounds totally fine and I agree that the English sounds bit weird - they've tried to do what is possible in Finnish but not in English. The Finnish sentence has an invisible 'se' before the verb 'haisee' - you usually cannot omit 3rd person pronouns nor demonstrative pronouns, but Finnish still tries to avoid repetition and if subject has already been mentioned once and there's already one verb before the second, you don't need to repeat the subject nor the pronoun, that is why the verb is there alone. In theory it's "Koira on likainen ja se haisee." and the better English translation would be "The dog is dirty and it stinks." because you can't omit 'it' here with English unlike you can do with Finnish. (At least that's how I feel about English! I'm not a native nor am I perfect at it.)
"Haista" is the infinitive, "haisee" its 3rd person singular.
Don't mix it up with the intransitive version: haistaa. In 3rd person singular that is haistaa. The difference is that haista means that something, in this case the dog, smells like something; and haistaa means that the subject itself is sensing the smell of something else:
- The dog smells bad. = Koira haisee pahalta.
- The dog smells a cat. = Koira haistaa kissan.