Player 1 builds from a city into a town and leaves one track coming out of the town. The next turn they do not extend that track. The rules state "if another player extends that unowned, unfinished track section they may claim ownership"
C = City, O = Town
C - - O -
Assuming player 1 controls the link between C and O, is there any way for player 2 to continue the unfinished track on the right of the Town?
You definitely can do this by urbanizing the town.
You can do this by building into the Town first from a city.
Let's say there is another city to the right of this city.
C - - O - C
I could not do this:
C - - O - - - C
because I would be building into already existing track.
Am I on the right track here?
Would it be a pain to add to this unowned track without using urbanization?
Since you can build out of any City, you could certainly do that. You just "start" the build at the City end and build to the abandoned track.