Response is just like with wired headphones - varies from model to model. Older bluetooth standards could add discernible compression artifacts to the sound, but newer standards and devices support AptX, which allows transmission of CD-quality sound.
Yes the wireless hop adds a slight big of delay. Usually around 20-50 ms. Your brain is pretty flexible and can compensate for about 250 ms audio delay before you start noticing lip-sync problems. But if you're into competitive gaming and rely on the audio for reaction time (your brain reacts faster to auditory stimuli than to visual), you should probably stick with wired headsets.
https://www.reddit.com/r/askscience/comments/2lfpyq/why_do_we_react_faster_to_auditory_stimuli_than/