Is there a minimum term for a WEEE authorised representative mandate in Germany?

Yes: three months, in writing and in German. § 8(1) ElektroG. You name the representative to the competent authority without delay and attach a copy of the mandate; the naming needs the authority's confirmation (§ 8(3)). A representative already holding 20 registrations needs an admission by the authority before more can be confirmed. When the mandate ends you report it, the naming ends once the authority confirms, and the representative remains responsible for obligations that arose during its term (§ 8(4)). ElektroG § 8