Recommended Post-op Care After Mole Removal?

I Don't recommend using hydrogen peroxide, it's irritating and toxic to cells. I tell my patients to wash normally with soap and water then apply simple moisturising creams. If I'm worried about hypertrophic scar (lumpy scars) I prescribe them silicon gel or sheets. 

You are on the mark…too much moisture (i.e. neosporin in this case) and hydrogen peroxide will definitely delay wound healing. Just think about getting a cut and leaving it under water all the time.Wait until you have some wound healing. The Scar gels and creams come more into play for scar formation, not wound healing. They do help, but aren't necessarily worth a great financial investment if things are progressing normally. 

Thank you for your question.  It sounds as if you have done all the right things.  Now at a week after the mole shaving, I would recommend you stop the hydrogen peroxide, neosporin, and bandaid, and instead apply Aquaphor or vaseline to the site several times a day.  The moisturized environment will allow the area to heal quickly.