Learn Croatian: Izbrisao bih tu objavu. - I would delete that post.

You use the perfective verb izbrisati because you are talking about a single, completed action: the post being fully deleted. If you used the imperfective brisati, it would sound like you would be in the ongoing process of erasing it or deleting things repeatedly.

Can I start the sentence with bih and say Bih izbrisao tu objavu?

No. The word bih is an enclitic, meaning it is an unstressed auxiliary word that strictly requires the second position in a clause. It can never be the first word. If you wanted to emphasize that you are doing it, you could add the pronoun to the first position: Ja bih izbrisao tu objavu.

I know bih forms the conditional mood, but what tense is the word itself?

The word bih is actually the first-person singular aorist form of the verb biti (to be). While the aorist is rarely used as a standalone past tense in modern spoken Croatian, its forms are kept alive largely for this purpose: acting as auxiliary verbs alongside the L-participle to create the conditional mood.

Is the word objava only used for social media posts?

While it is the standard word for a social media post today, its broader traditional meaning is an announcement, publication, or declaration. It is derived from the verb objaviti, which means to announce, declare, or publish something.