What effect does peroxide rinse have on teeth and mouth?

Not much. Hydrogen peroxide (3%) is approved for use as a mouthwash when diluted 1:1 with water. This very weak solution may fizz slightly as it liberates oxygen, but this is unlikely to be of great value. It is not a substitute for good flossing or brushing. It is unlikely to any meaningful whitening effect. In short, its value is limited.
Mostly cleansing. Peroxide rinse is effective against many bacteria and is commonly used. If used in too much concentration or for too long it can chemically burn the tissues, otherwise it is a great cleanser. It can alter taste as well. Peroxide does create free radicals and some worry about cancer from that. Enamel on teeth is cleaned and whitened slightly by peroxide rinse (not an effective whitening method).
Not a good thing to. Probably harming your good antibodies and salivary enzymes than benefiting. Stop doing that this won't have any whitening effect.
Not good. I don't recommend peroxide because it oxidizes causing tissue damage. It also kills the "good" bacteria in your mouth. I highly advise against rinsing with peroxide.
Minimal if. Peroxide is really not a great disinfectant in otc solutions it is better than some. Actually in low concentrations can be used as mouth rinse, but it really should not cause much prob. If concentration is high enough it can be caustic and cause blanching and possible burning sensations. Hydrogen peroxide is used for bleaching teeth and it burns.