Does the coxswain of a Masters crew have to be a Master, and is there a minimum weight?

No age requirement. A Masters crew must be made up entirely of Masters rowers, but the coxswain doesn't have to be a Master and can cox any Masters event regardless of age (4-104.1(d), 4-103(f)). The coxswain's age isn't counted in the crew's average.

Any sex. A coxswain of any sex can cox men's, women's or mixed crews (4-105).

Minimum weight (4-105.1):

A coxswain under the minimum carries deadweight, placed as close to their torso as possible and kept in the boat for the whole race. If the deadweight isn't in the boat at the finish, the crew isn't placed. Coxswains weigh in once a day, in racing uniform without shoes. Not weighing in at all makes the coxswain ineligible.