I want to batten the walls and install some insulation, but I want to
make sure it's rodent resistant/proof.

Probably use exterior grade osb on the inside, but what insulation to
use ? rockwool or Glass fibre mat ?

What would mice etc not like to nest in is probably the query.

Also need to insulate an up and over garage door, what's best to stick
on here. Clearly I won't be skinning it over with wood due to excess
weight. How about around the edges where the gaps are ?

Thanks

Paul.