This summary is drawn from the Rebbe's letters to him (about 5709–5722, 1949–1962); I did not find a published biography.
Scholarly questions. Much of the correspondence answers his Torah questions: the source of the saying that Eretz Yisrael will expand to all lands (IK 2225); why Shavuos is called "Zman Matan Torah" even when its date does not match the day the Torah was given (IK 6301); clarifications of Zohar passages (IK 6744, 6919); and reading "Vayechal" on fast days, following the ruling of the Tzemach Tzedek (IK 7208). On cemetery customs, the Rebbe advised consulting the local rabbinic authorities (IK 7675).
The role of a Rav. The Rebbe taught him that a Rav's influence must reach beyond personal observance, actively spreading Judaism and Chassidus (IK 8135), and encouraged increasing joy in Adar and spreading Chassidus both within oneself and outward (IK 6744).
Sources: the Rebbe's letters cited above.