Power Wemos D1 mini from battery
According to the (RT9013 schematic), it has an operating voltage of
  2.2V to 5.5V. This I do not understand, I thought the LDO could only drop to a lower voltage?


Yes, a linear regulator only drops voltage indeed.

The operating voltage range here refers to the INPUT supply (before the drop). However, this wide input range refers to all the devices in the family (fixed output). Some of them can output as low as 1.2 V, but will still need a minimum of 2.2 V, which is 4 times the +250 mV rated dropout and 2 times the + 0.5 V at which the electrical characteristics of the datasheet are rated. 

In your case, you'll pick the 3.3 V out version, so don't worry about this spec.


  I've also read that the lowest the battery get is 1.2V or so, which
  would mean that the total voltage would only be 3.6. This is still
  above the 3.55V minimum, so it should be OK?


It should be OK, but with no margin whatsoever. Any small drop in voltage would be passed to the output, and some of the specs in the datasheet may not be fully met (they're rated at +0.5 V). Do you actually foresee to draw 500 mA?

One thing you must take into account is the fact that the maximum dropout at 500 mA is 400 mV over the full operating temperature range. At 500 mA the regulator will start to raise its temperature, and the dropout performance will also degrade up to the point where it doesn't regulate anymore. See curves below: