Learn Finnish: Kun sota alkoi, silta olisi tuhottu. - When the war started, the bridge would have been destroyed.

This is the conditional perfect passive, which translates to 'would have been done'. It always uses olisi as the helping verb, followed by the passive past participle of the main verb. Here, that is tuhottu (destroyed).

Why is the word for bridge (silta) in the basic nominative case instead of an object case?

Because the second clause is in the passive voice. In Finnish, when an action is done to a whole object in a passive sentence, that object takes the basic nominative case (silta) instead of the genitive-like accusative.

Shouldn't both halves of a conditional sentence use the conditional mood? Why is alkoi just in the regular past tense?

If this were an 'if' sentence, both sides would use the conditional mood. But this sentence uses kun (when). It states a fact: the war actually started. Only the destruction of the bridge is hypothetical, so only that half gets the conditional olisi.

How is the form tuhottu built from the basic verb tuhota (to destroy)?

Tuhota is a Type 4 verb. To form the passive past participle for Type 4 verbs, you generally replace the final -a or -ä of the infinitive with -ttu or -tty. This easily gives us tuhottu.