I would say, No, No.
For the first one, at my job we are not impressed with people contacting us unless we have called them. You got an interview, now wait for either a rejection or a job offer. (Except for an interview thank-you letter.)
For the second one, it would put your old professor in an awkward spot and how you did in a class isn't really an indication of how you will do at a job. Plus, I imagine the professor's spouse would be VERY taken aback if he were to, out of the blue, start talking about an applicant for a position at her job. That would just be...weird. At my job, recruitment is pretty confidential so we are expected to not blab about applicants.