Is my iMac 27-late 2012 really obsolete?
This one is almost five years older than yours and it runs perfectly ok:
El Capitan wasn't coping well with the original 4 GB. Adding another 2 GB extended its useful life a number of years. But if your Mac is running well with what it has there is no need to change a thing.
Its original spinning hard disk drive is still going strong, but I installed additional memory a while ago (which is not a supported configuration). Replacing the hard disk drive with a solid state drive would increase its performance even more, but since it runs well enough with the original HDD I haven't bothered (yet).
I'm guessing the processor is the issue.
Not exactly. I surmise the messages you are getting address Apple's decision to abandon support for 32-bit apps and processes; a decision they made about a decade ago. If that does not describe the warnings you're encountering, describe what they are.