The salt does not dissolve into the oil, it contains it like a glass contains water.

Salt caverns are very interesting for storing all kinds of oil / hydrocarbons / chemicals. Most fluids found naturally in the ground (oil, natural gas, groundwater) are stored in porous rock. Rocks like sandstone seem quite solid on a human scale, but do have microscopic voids which are filled with whatever kind of liquid/gas in their natural state. We drill into this rock and the liquid/gas slowly moves through the pores of the rock and come up the well. After the liquid/gas is removed, the rock still remains, there's not like a giant cavern you could stand in remaining underground...it's just still solid sandstone.

Sometimes we can re-use these sandstone/limestone reservoirs after extracting the oil from them. Natural gas storage facilities sometimes set up shop and pump surge/excess natural gas into the empty oil fields in the off season (summer for northern states) and then extract it again in the high season (winter for northern states to heat homes). However, due to these being natural formations, sometimes there are discontinuities in the boundary layers of impermeable rock so that some of the gas leaks out of the reservoir over time. The storage facility says basically "Anyone who can deliver natural gas to this pipeline may store their natural gas with us. We will mix it into the formation with everyone else's gas, and then pump it out later. For every 1000ft^3 you give us, we will give you 900ft^3 back, and charge you $4 for the privilege."

Salt caverns are very different! They're not porous. underground there can be immense - up to 2 miles long by 2.5 miles wide, by 1800 feet in depth. Sometimes miners work the salt face like coal mines, but because salt is soluble in water, sometimes it is mined by drilling a small hole into it, then pumping water through the hole and bringing saltwater back up (typically this is the method that chemical plants such as the one in Freeport, TX use to obtain salt/chlorine to make PVC/etc).

After washing out a lot of salt using water, there is left an enormous void - a room large enough to park many vehicles. It takes a lot of engineering to make sure you washed out the cavern into a shape that is structurally sound. If geologists determine that the cavern is structurally sound, then they can try filling it with hydrocarbons.

Unlike sandstone/limestone formations, salt caverns rarely leak much product out of storage, but the salt is rarely 100% pure (92-98% is common) so some of the other minerals can dissolve into the stored product and needs to be removed...but typically the cost of cleaning the stored product is minimal compared to its value.

To answer your military question: Yes and no. Yes, all storage facilities of strategic value are mapped by all world powers and targeted in a WW3 scenario. But you wouldn't generally try to cause an explosion inside one. In fact, a stuxnet-style attack would be somewhat unlikely to be able to, as there's generally no oxygen/air inside the storage caverns. The oil is removed by pumping water below it.

A more likely attack would be a conventional or nuclear explosion above ground, knocking out the equipment/pipes needed to extract the oil from the cavern.

Other targets of similar or even greater value would include any locations where multiple strategic oil pipelines cross eachother and exchange material flows (e.g. Hattiesburg, MS), or where the refining takes place (e.g. Houston, TX). Even just one pipeline not working unexpectedly can cause noticeable fuel disruptions/shortages, such as when Colonial Pipeline had some accidents in 2016, parts of New England area faced fuel shortages.

Also, we have plenty of above-ground oil storage (in typical tanks) in addition to underground storage. However, the strategic petroleum reserves (800 million barrels) are truly massive and dwarf the entire private industry storage in USA (above and belowground combine to 650 million barrels).