Hi Karen,
Quite the project you're embarking on. Certainly information regarding your unit is important. The links posted didn't "link" really at least for me. However, to address a couple things based on your descriptions seems easy enough.
1) That rusty looking material in the core of the steel cable more than likely is the hemp core, that typically provides lubrication to the surrounding strands of the wire rope, so yes it'd look like a fibrous string thing.
2) The broken wires on the steel cable itself. Certainly there are specs for maintenance and life expectancy values, depending on their ratings by the manufacturer. I'd replace them like for like, based on the breaks you've already seen.
3) Maintenance. Anything that slides, glides, spins or any other moving rotating moving parts will need attention at some point in its life. Electrical contactors/relays included. Obviously sealed bearings are what they are, but look for any type of zerk fitting for grease, gliding chair rails need lubrication (grease possibly) or you may have a trolley type wheel, etc.
The steel suspension cable termination points are very important, but how to make a suggestion here I won't, but will tell you to seek information via the Internet about your brand and model for technical help, but it may be self evident to an elevator trained person, but search it out thoroughly as this is of major import...the suspension means of the chair and counterweight if present.
Continue your search for help, and more than likely others here on the forum will want to help as there an awful lot of smarter fellas than myself here.
Good luck on your project, and yes, "Safe" on your project is of paramount concerns.