A more expensive microphone will include a calibration graph guaranteeing the response. Professional acousticians in labs making absolute measurements might benefit from that, but most of us measuring rooms don't need a guarantee of accuracy.
Calibrated means someone actually measured that specific microphone, and again such microphones come with a printout. I have an old AKG C451 with a CK22 omni capsule I bought for room testing in the late 1970s. The capsule came with a printed calibration graph.
I don't know any easy way to calibrate microphones at home, and again it's not really needed just to see how much better your room is after adding bass traps.

Yes, a measuring microphone is excellent for recording. I used my DPA 4090 for every track of my elaborate Tele-Vision video. (Other than the few tracks that were played from samples.)
--Ethan