Can anyone rent a car in Hungary and drive through Zagreb, Split, and Dubrovnik? Can the car rented in the middle of Bosnia and Herzegovina pass through? Thank you for experienced friends to inform me ?

I don't know if you can rent a car in Hungary, but you can rent a car in Belgrade. The section between Dubrovnik and Bosnia and Herzegovina has left an outlet to Bosnia and Herzegovina, so you need multiple Schengen visas or to circumnavigate the island.

Whether the car hire company you choose allows crossing Bosnia and Herzegovina

No problem. When I went to Du City last year, there was a small blue house checkpoint on the way to Bosnia and Herzegovina, which was similar to a toll booth. I just handed over my passport and got a stamp.

The car I rented from Budget is okay. When picking up the car, I said I would cross several countries, and I just needed to pay the cross-border fee. I crossed 5 countries last year and paid 88 euros for the cross-border fee.

Just opened last year, you need to explain to the car dealership when picking up the car, and then just pay a little extra money. It's just a few tens of euros, I remember.

Can pass, there was no border control when entering.