To do that, you need to expend roughly the same amount of energy that is needed to first liquify and then solidify the oxygen.

In fancy chemistry-speak it's called "entropic loss". You do gain some of that energy back when the battery is charged, as oxygen goes from a well-ordered solid state into the gaseous state. But it's not 100%.

That claim is based on dividing the stored energy by the mass of lithium, which is incorrect.

The product of the reaction, i.e. lithium oxide, is stored in the battery, so a lithium-air battery can never be lighter than the lithium oxide.

Because the mass of lithium oxide is what counts, the energy per mass of pure lithium, which is indeed not much less than for gasoline, must be divided by a factor that varies between 2.14 and 5.57, depending on the construction of the lithium-air battery.

The best value of 2.14 is when the discharged battery contains only Li2O. The worst value of 5.57 is when the discharged battery contains only lithium superoxide, LiO2.

In the parent article, they claim that their discharged battery contains mostly Li2O, with only small quantities of peroxide Li2O2 and superoxide LiO2, but the exact amounts of peroxide and superoxide have not been measured.

So when computing correctly the energy per mass ratio, for lithium-air batteries it is limited to a value less than half of that for hydrocarbons. In practice batteries need a lot of materials besides the active reactants, so the achievable energy per mass ratio will be several times lower.

The advantage of hydrocarbons, regardless whether they are used in living cells, thermal engines or fuel cells, is that their reaction products are eliminated into the atmosphere, so their mass does not matter. The energy per mass for carbon atoms in hydrocarbons and for lithium atoms in lithium metal is approximately the same, but with lithium it is impossible to neglect the mass of the oxidant, like with carbon, because the reaction products cannot be dumped outside.

So for any battery except for fuel cells, what counts is the sum of the masses of the reactants, e.g. lithium + oxygen in the best case, or e.g. zinc + manganese in the cheap non-rechargeable batteries. It is wrong to compute the minimum mass of a battery by using only the mass of one of the reactants, like in the parent article, instead of both masses.