What language family does Hungarian officially belong to?
Finno-Ugric, of the Uralic language family. Specifically, Magyar is assigned to the Ugric branch (whose closest language kin are Khanty and Mansi) . While the classification has been controversial since its inception a century and a half ago, official etymologists and linguists in Hungary do not receive support to study connections to languages outside of those considered "official."
What language family does Lepcha officially belong to?
Sino-Tibetan. Specifically, the Tibeto-Burman branch.
Could Hungarian and Lepcha be related?
It is quite possible. Both Hungarian and Lepcha have had issues surrounding the classification and dating of their respective language families. However, since the 19th century, alternative theories regarding Hungarian language kin have included Lepcha.