Thanks for posting your query at DoctorSpring.com. I can understand your concern. In consult with the team Dermatologist, Dr Kruti B -
Moles are diagnosed as malignant based on the abcde criteria. One of the criteria is colour variegation I.e the reddish tone you are mentioning. However it is possible for a non cancerous mole to get irritated due to friction like it may be in your case. You can put neosporin and cortisone cream over it for five days and see if it helps.
Biopsy usually takes a couple of weeks, it is the method of diagnosis. Initiating a treatment in melanoma as soon as possible is important however, another doctor may also take as long to give the results, you can find out, and if the biopsy results are taking long you can request your doctor for a dermatoscopy where they see the lesion through an instrument and a melanoma can be detected.
Please let me know if you have any queries.