Mulligans, including mulligan putts, reduce ball durability by -2 hits for each mulligan used.
Shots made from fringe are approach shots, even if you use your putter, and reduce ball durability. By the same token if you use your putter from fairway it counts as an approach shot and ball durability is reduced.
I suspect, but don't know for sure, that any shot made on the green, regardless the club used, counts as a putt and does not reduce ball durability.