Best way to power an ESP8266 (3,3V) from AA batteries with LDO
I see a lot of projects using the MCP1700 LDO in combination with the
  ESP. This LDO has a very low quiescent current (&lt;2 µA), but the
  datasheet says the max output current is 200mA, from what I've read
  the ESP can have peaks up to 250+ mA. Isn't this a problem?


It may not be a problem if the output capacitor on the regulator can provide the 50 mA extra over a short period of time without drooping the output voltage too much. If the capacitor is (say) 100 uF and the current demand is for (say) 0.1ms you can use the formula I = C dv/dt to calculate the voltage droop (dv) in the 1ms (dt).

So 0.05 amps x 0.1ms = 100 uF x dv, hence dv = 50 mV i.e. not much of a droop. Having said all this I don't know how long the peak of 250 mA will last for.


  Which LDO should I use to have a stable but power saving setup?


I have no idea what you ask?


  How can I calculate the time this setup will work until the batteries
  voltage drops below their minimum?


If each battery is rated at 1900 mAh, the total series connection will still be 1900 mAh but that rating may be a full depletion down to 2.5 volts (just an example) so play safe and assume you only have about 1000 mAh. Alternatively read the data sheet for the battery type and see what graphs it shows. If it's a decent battery it will have data sheets. If you bought it from ebay or aliexpress then don't count it it.


  Which caps should I use for decoupling the LDO? Just the ones
  suggested in the datasheet? Is it important if the capacitors are
  ceramic or electrolytic?


Use the types in the data sheet but bear in mind you should try and use low ESR electrolytics unless the DS specifies otherwise.