Is there any difference between the following sentences?
Would you mind if I smoke a cigarette outside?
Would you mind me smoking a cigarette outside?
Would you mind if I prepare a tea cup for you?
Would you mind me preparing a tea cup for you?
To me, "Would you mind me [doing something]?" and "Would you mind if I [do something]?" seem equivalent ways to ask before doing something, but maybe there are differences between those phrases that a native speaker would catch.