Answer by Will Ness for Haskell: tail recursion version of depth of binary tree

to your 3., yes, e.g. by use of CPS technique (as shown in Chris's answer);

to your 4., correct.

to your 2., with lazy corecursive breadth-first tree traversal we naturally get a solution similar to Chris's last (i.e. his #5., depth-first traversal with explicated stack), even without any calls to max:

treedepth :: Tree a -> Int
treedepth tree = fst $ last queue
  where
    queue = (0,tree) : gen 1 queue

    gen  0   p                     = []
    gen len ((d,Empty)        : p) =                     gen (len-1) p 
    gen len ((d,Branch _ l r) : p) = (d+1,l) : (d+1,r) : gen (len+1) p 

Though both variants have space complexity of O(n) in the worst case, the worst cases themselves are different, and opposite to each other: the most degenerate trees are the worst case for depth-first traversal (DFT) and the best case (space-wise) for breadth-first (BFT); and similarly the most balanced trees are the best case for DFT and the worst for BFT.