You must not ride it much. Chains and gears are wear items on bicycles. They should be cleaned and lubricated regularly.

The gears and chain generally wear at the same rate. If you put a new chain on a worn out old set of gears they will likely slip.

The chain wears out muuccchhh faster than gears when properly maintained. If you let your chain stretch too much, then you'll wear out gears quickly. This isn't them wearing out at the same rate, it's a worn out chain ruining your cassette and chainrings.

As a rule of thumb, one should replace the chain every 1000 to 5000 km, depending on the force you exert when cycling. Since the chain otherwise tends to stretch, it will eventually be misaligned with the chain wheels and start to wear those down. As the latter are more expensive to replace, it’s a good idea to replace the chain first.

Not really. All the parts that would be damaged by that are designed to be replaced. My girlfriend did the same to her bike not too long ago. I spent about $80 and an afternoon to get it back in shape.

I've never tried to find parts for a bike that old before, so that might be the challenge. I did have a late 70's Schwinn that I converted to a fixie and had no issues there, but that was a much more invasive job. At the end of the day, I'd replaced almost everything on that frame, so I had less compatibility concerns.

Take it in to your local bike shop if you are really worried. The biggest factor in your chain's health (assuming you have kept it in a nice dry place over the years) is how much you ride it. It's a mechanical part, so it shouldn't simply go bad in an inert state. If you are riding it regularly though take it to your local shop and have it looked over. A new chain w/ installation should only run you about 20 or so bucks for a low end chain.

Simply spraying some lubricant (like Teflon spray for example) onto the chain, running through all the gear combinations, and wiping the excess lubricant off the chain every couple months goes a long way. It helps your bike make less noise, extends the life of your chain, and helps keep your chain from turning totally black with grease. (Which means you're less likely to get grease stains on your pants when you ride.)

The chains stretch over time, and teeth of the cassette get worn. The more you ride the quicker this happens. If you take it to a shop, they can quickly measure the chain to make sure you're good. Chains are pretty common to replace and shouldn't be too expensive. Cassettes as well, they're often changed at the same time as the chain. Otherwise, regular cleaning and lubing will help extend the life.