In the salt it's pretty similar. The salt caverns are basically just like the conditions where the early, easily-exploitable oil deposits ("gushers") were.

No, it's because gasoline is not a very thoroughly refined chemical, so it basically has a bunch of aromatic rings, benzene, and all sorts of monomers floating around, that rapidly (~year or two) turn into varnish and other polymerized nastiness when they oxidize and crosslink. I believe gasoline was originally a refinery byproduct that was used because it was so cheap as a fuel.

Diesel and other heavier fuels generally are much closer to aliphatic hydrocarbons (straight chains, not aromatic) and so are much more storable, because aliphatic hydrocarbons have much less opportunity for oxidation (they're like saturated fat, i.e. much more stable).

You can get polymerization inhibitors to put into gasoline if you want it for long term storage, with the caveat that they produce some nastiness at the bottom of tanks that you'll have to avoid when pumping it out, and I think they can also slightly reduce the octane rating.