How does your breathing change during rem?

Yes REM matters. When you switch into REM sleep, your breathing becomes more rapid, irregular, and shallow. During REM sleep, breathing is erratic, with variable respiratory rate and tidal volume and frequent central apneas. Thus, breathing is impaired during sleep compared with wakefulness.
It doesn't. Rem is experienced while sleeping. Patients with apnea have problems achieving REM stage of sleep. Breathing does not affect rem.