First off, what exactly is a checksum on a calculator? What does it check and how does it work?

And, can it be calculated using Axe? If so, how exactly would I perform a checksum on all programs in either RAM or Archive, but only the programs and nothing else?

Then, how would you perform a checksum on the operating system?

**I would like to do this in Axe, if possible.

And what should the sum be? How do I use it to decide if something is not right?

Ok. What sort of errors would your above method check for?

And can the OS itself be checksummed in Axe, or would that need asm?

Typically transmission errors (TI's transfer protocol uses it).
It can as soon as the degree sign (for address-of) becomes official. (You have to be able to select an arbitrary page, which can be done by manually setting a "file pointer")

Could someone, if they have the time, write an asm routine that calculates a simple checksum of the OS and saves it to an appvar, "BChkSav", overwriting an old one if it is found. If it isn't too much of a problem. It might be an interesting addition to a secret program I am working on.

Your better off with something like an MD5 hash, as a checksum could be easily foiled by adding an extra word somewhere so the checksums align themselves.

MD5... that seems to be a bit too much
It's another hash function, and has some desirable features including the difficulty of making something with a certain hash.
Only problem is computing it will take a bit longer than you would probably like (probably on the scale of a couple programs a second) because it involves 32-bit calculations, which are more difficult in Axe.
However, MD5 is not very secure (a few seconds on a modern PC to break it), and moving to SHA-2 is, best put, ridiculous.
It mainly depends on the protection you want. If you want more than a simple checksum or variations thereof, I can only think of MD5...
Somebody else can probably tell you more than I have
Link to documentation on MD5

Badly written is easy. Malicious is what's difficult.
For badly written, a checksum is more than enough. You just need to increase the size of the checksum to increase security (Though only 2 bytes should be necessary anyway)
Malicious is harder, but by increasing the checksum size to, say, 4 bytes instead of 2, it becomes harder for a program to change the OS without changing the checksum.
I'll write some ASM to create a 2-byte checksum of the OS and return it in hl (for Axe purposes).
* calcdude goes and codes

Any change to the OS good or bad would change the checksum. Also, I wouldn't worry about accidental changes to the OS because that is close to impossible to do by accident. Besides, the ones that happen by accident require an OS reinstall because the calculator would not even be able to boot.