How is airspace in the Persian Gulf determined?
The map you see is for Flight Information Regions (FIR's), that's different from sovereign airspaces.
Sovereign airspace by international law "corresponds with the maritime definition of territorial waters as being 12 nautical miles out from a nation's coastline. Airspace not within any country's territorial limit is considered international, analogous to the "high seas" in maritime law."
Can military planes fly across this "international airspace"? Yes. An example is Russian war planes flying very close to the UK airspace, passing through many FIR's.
An example for a disagreement is between Greece and Turkey, because they have so many islands between them. According to some sources just last month Turkey violated Greek airspace 141 times in one day. The point is, there are disputes over sovereign airspace just like with land borders.
Why, for instance, does Bahrain have such a disproportionately large amount of
airspace [FIR] for its size?
It's a matter of "administrative convenience". Both Bahrain and Qatar gained their independence from the UK in 1971. It's quite likely the current FIR shapes were carried over from before and based on where the radars were initially installed.
[...] Qatari flights fly over Das island, despite it being part of UAE territory.
Checking the official Qatari aeronautical chart above, the Emirati islands are not part of Qatari airspace.
See below the official FIR chart for Bahrain, despite covering the area north of Qatar, airplanes in and out of Qatar still fly north.
To illustrate the operational and administrative convenience I've split the commonly traveled east-west airways between Saudi Arabia, Bahrain, Qatar, and UAE. Now you can imagine the impact it will have on ATC, flight planning, flying, etc.
To further illustrate the region, abeam the four countries the distance is a mere 150 NM, or 20 minutes in cruise. If the FIR is split, a single flight would need to contact 4 different frequencies in such a short time. And if that flight is descending/climbing to/from one of the four countries, multiplied by the amount of flights, then you can imagine the substantially higher workload on both ATC and pilots with regard to coordination and hand-offs.
A similar oversized FIR example is Singapore FIR (shown below).