TSP can be used to solve “Hamiltonian path”: is there a path connecting all nodes?
You do this by specifying a TSP instance where distance = 1 if two nodes are connected, distance = H otherwise (where H is some huge number), and checking if the shortest path has length n (instead of n + H - 1)
An approximation cannot be good enough to solve Hamiltonian path.
I have never seen this mentioned, but I could imagine that "almost Hamiltonian path" where you are supposed to find a sequence of n nodes of which at least n-m are connected would be NP-complete for some combinations of n and m, for example if m=1. In that case, an approximation better than m * H would solve the "almost Hamiltonian path" problem.
You can find a path by starting at some node and then always going to the nearest neighbour. You can take that as a baseline, or n times the longest distance, or the sum of the longest distances from each node. I suspect you won't be better than this baseline times some small constant.
On the other hand, you can turn a TSP into TSP with metric by increasing all distances by some value. You can determine the minimum increase to guarantee triangular equation, and go from there.