Learn Croatian: Bio bih tražio anesteziju za taj karijes. - I would have asked for anesthesia for that cavity.

This is the Past Conditional (also called Conditional II). It is formed with the past participle of biti (bio), the conditional helper word (bih), and the past participle of your main verb (tražio). While you need to recognize this for reading and formal writing, in everyday spoken Croatian people often just use the regular conditional (tražio bih) and rely on the context to show they mean the past.

If I wanted to start the sentence with tražio, how would the word order change?

The word bih is an enclitic, meaning it naturally sits in the second position of the sentence. If you move tražio to the front, bih must immediately follow it, pushing bio to the third spot. The sentence would become: Tražio bih bio anesteziju za taj karijes.

Why does taj karijes look like it is in the nominative (dictionary) case even though it comes after the preposition za?

The preposition za (when it means "for") requires the accusative case. The word karijes (cavity or tooth decay) is a masculine noun. Because a cavity is inanimate (not alive), its accusative form is exactly the same as its nominative form. That is why the pronoun taj stays as taj instead of changing to tog (which you would use for a living masculine noun).

I usually translate as "to look for". Does it also mean "to ask for"?