Tires do age out -- is your spare a full size / matching wheel style to the other four? If not, it'd look weird to rotate it and I'd probably just throw a cheap tire on there for emergencies. In the event you actually have to use your spare, a 10-12yo tire is likely to be unsafe. You can inspect it for dry rotting / cracks and maybe stretch it out a bit longer, but I do prefer to err on the side of caution with tires.
I don't think there's a way to check battery life on TPMS sensors. It is definitely not a DIY job, as you have to dismount the tire, install the sensor, remount the tire, and then re-balance the wheel/tire assembly. I would not go cheap with the sensors (I use Denso myself), just because I'd rather not risk having them die years earlier than they should (I have been buying Denso for my Lexus/Toyota vehicle). I also replace them when I get new tires, unless I already know the age and they likely have several years left.
If you're due for an overall tire replacement soon, I would just replace them then and check your tire pressures every time you put gas in the car. TPMS is superior than checking manually, but if you're close you can usually get them installed at minimal / no extra labor charge during a tire replacement.