Depots can only be built on tracks between two cities
Can you cite your reasoning? The rulebook only states:
"Take the depot from your supply and place it on any
railway space that doesn't already have a depot on it." (Publisher's emphasis, not mine)
So it seems that the rail lines that run off of the board are still okay, since they still consist of railway spaces and no other limitations are stated.
As for the partial hexes, I would point out that a depot on a partial hex is nearly useless, since the rules are clear about the fact that your airship cannot end its movement there. And if you're only building the depot for the points, then there are probably lots of other legal placement options.