This is not hard, and for 8km height you could manufacture them for under $2k/piece. Even if they'd be made to work with hydrogen, as that raises the demands on the construction. If you'd find the FCC to be willing to allow something like this and are able to handle both thunderstorms and just the general wind load on the tether, this should not be hard.
[0]: https://en.wikipedia.org/wiki/Single-wire_transmission_line
Regarding the damage from it when falling down, you have to remember that each fiber weighs 50g/km, and the wire is also below 1kg/km, even for rather low heights. This should not be a problem, considering the ease of breaking an optical fiber optic at any remotely sharp edge. A small bird (the size your cat can catch) could probably snap it if he hit's it with some force.
1 kg/km • 50 km • 9.8 m•s^-2 • 50 km = 24.5 MJ
10 MJ = kinetic energy of the armor-piercing round fired by the assault guns of the ISU-152 tank
I think that's just about enough energy to cut just about any living thing in half. Maybe it work if the cable was made of spider silk or Kevlar.