Your skill will accomplish
what the force of many cannot
How do you define an orbit for property purposes?
I'm sure there are many definitions. I'm also pretty sure that orbits are already defined in some fashion because there are thousands of satellites up there that (IIRC) have never collided in the 50 years we've been putting stuff up there. I'm pretty sure that a company devoted to registering orbits would come up with a pretty good way of describing them.
That there are important orbits that may have many things in them. (Geostationary orbit is the most notable example.)
Said company(s) would start their registries non-empty.
That orbital mechanics are chaotic, so any object placed in orbit will eventually wind up in a different orbit.
Either that can be accounted for in a satellite without the ability to maneuver or maneuvering capability would be required.
For each owner of an orbit, it is easier to just avoid stuff crossing your orbit than it is to clean that stuff up.
If you don't want to clean it up and feel you can avoid it, then great! I've gone without health insurance for months at a time in the past. Even though I was fine, I still pay through the nose right now for it. If I just paid M$50 to make a satellite and get it into the right orbit and I have B$20 in future revenues riding on it, I'll be willing to pay M$10 (or something in that range) to increase its chances of survival. It's almost like a company that pays for improvements in a CPAN module. The company is paying for the benefits they are reaping and costs are calculated from that perspective. That there is an improvement to a commonly-held property is good advertising.
Also any attempt to clean up junk that is crossing one orbit is fairly likely to move it into someone else's orbit.
I think BrowserUk's solution would work quite nicely. With registration of lower orbits, that kind of work is pretty simple to time.
Even if you propose a solution to these problems, you have the bigger problem of how to agree on regulations that have to cover multiple countries. Do you really think that China or India is in any mood to accept a division imposed by the USA, Europe and Russia about who may use what parts of space? And what about the smaller powers? According to Wikipedia there are 45 countries with space agencies. Many, admittedly, do not launch their own satellites. But getting that many countries to agree on a complex legal system involving space is going to be very, very tricky.
What legal system? The only thing that even touches legalities is what courts to use for civil suits. Multinationals have been dealing with that issue since well before the East India Tea Company. Forum-shopping is a completely normal part of corporate law. If there's a crime on board a ship (or other space entity), you apply the provisions currently attaining to a crime at sea. That's worked well enough for hundreds of years. The key here is to avoid the nation-states and to appeal to the corporations. Yes, corporations are rapaciously greedy bastards that look out only for number 1. I can't think of many countries that don't qualify for that description.
I don't have all the answers off the top of my head. I do believe, very strongly, that all the issues you're raising are issues that are solvable by for-profit entities. And, frankly, solvable in better ways than nations could do so. In my eye, the "Tragedy of the Commons" is a for-profit opportunity to take the commons private for cheap, invest to improve it, then lease out usage. At least under private management, the commons wouldn't deteriorate because the owners have a motive to keep it in good shape.
My criteria for good software: