Learn Croatian: Stari rudar tražio je zlato jer je rudnik bio prazan. - The old miner searched for gold because the mine was empty.

Past verbal adverbs are almost exclusively formed from perfective verbs. Since tražiti (to search) is an imperfective verb, you cannot use it here. You would first need a perfective equivalent, like potražiti, which gives you potraživši zlato (having searched for gold).

What if I wanted to use a past verbal adverb to shorten "because the mine was empty" to "the mine having been empty"?

The verb biti (to be) does have a past verbal adverb form: bivši. However, using bivši in a sentence like this sounds incredibly archaic and formal in modern Croatian. It is much more natural to keep the standard clause using jer (because) or budući da (since), exactly as written here.

Why is the adjective stari (old) in its definite (long) form, rather than the indefinite star?

We use the definite form stari because we are talking about a specific, known miner ("the old miner"). If we were introducing him for the first time as "an old miner", we would use the indefinite, short form star rudar.

Since we used the definite form for "stari", why is the adjective prazan (empty) in its indefinite (short) form?