Yes. Though not as fast as a tire out in the sun. Tires in the sun will suffer UV damage as a primary form of degradation (obviously mechanical were too if in use). Spairs don't see the sun so not a problem, but they do oxidize. Ozone will attack the rubber in the tire and cause it to go brittle. that is what causes all the tiny cracking you see on old tires. Generally it is recommended to replace any tire over ten years regardless of use. Interestingly this applies to brand new tires that have been sitting in a warehouse. It is worth learning how to read a tire date code so you can check the tire age. This helps with buying used cars, making sure a tire company didn't sell you NOS tires, and just general safety.

https://www.tirerack.com/upgrade-garage/how-do-i-read-a-dot-tire-identification-number-tin

For you I would just swap it out next time you buy tires. If it is a full sized spare then just save the best of the tires you are replacing as the spare. Save you buying a whole new tire that you will probably never use.

Yes, a good mechanic cycles the tires every service, and recommends switching all 5 before end of life. I've had better luck though convincing my clients to start changing tires one tire at a time starting 1.5 years before end of life at every service. Since a lot of clients don't like shelling money for all 5 at the same time.

It's been 9 years since I've been a mechanic, so take my advice for what you will.

Yes, it needs to be replaced at this point. A spare will last a little longer than regular tire as long as it's protected from the sun, but after 10 years, it's time to replace it.

Yes.

Not a mechanic but even if it is technically too old you maybe be fine if you keep it and in the event you get a flat simply replace the flat with a new tire, after getting to a tire shop on the spare.