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I guess I don't understand why you're (apparently?) more worried about the rotor groove and the squealing than the car's pulling to one side while braking. The latter is a serious safety issue. I'd fix the pulling first and worry about the other stuff second. I'm not sure the groove is a serious problem as long as it's not getting bigger (think slotted rotors).
This could potentially be a sign of a sticking or seized brake caliper qhich in turn will not only cause the noise being heard but will also cause the vehicle to pull while applying the brake. This will also allow the caliper to groove the rotor in what appears to look similar to a vinyl record. I think the best answer would be to get the brake system diagnosed so that further damage and expense ia jot accumulated and/or the brake system does not fail causing an accident or much worse. I can almost bet once diagnosed and repaired the pull will no longer be present.
Road debris kicked up when the tires roll over will sometimes get stuck between the pad and rotor and will scour the rotor. Same thing happened to me right after I did a brake job with new pads and rotors. Other than cosmetic eye-sour, functionally it doesn't seem to affect braking.