The “transmission” is called that because it “transmits” the power of the engine to the wheels.
You can think of the CV axle as the line that hooks the transmission to the wheels. So it isn’t part of “the transmission” but part of the action of “transmitting.” The engine spins. That spins the transmission. The axle connects the transmission to the wheels so they spin too. (Not all axles do that - some are just passive axles and are sort of just pipes that the wheels spin on. But they can’t transmit any power).
The differential is basically part of that link. The problem is that whenever you go around a corner the wheels on each side of the car have to spin at different speeds. (The outside wheel actually has to spin faster than the inside wheel). The differential is basically some gears that allow that to happen.
You can have a differential that is outside of the transmission. But they are sometimes built into the unit called the transmission. So in technical terms a transmission is always different from the differential. But in practical terms, as on your car, it can all be integrated into the same hunk of metal.
Explaining is a CVT is partly beyond what I could or would type up (could take a book).
There are various websites that tell you “how things work” and many of them are great. They can include simple descriptions with diagrams, some of which can be animated. You can get a clear picture of how transmissions work, including CVTs and how differentials work etc.
Normally I would tell you to find your best local transmission shop and take the car to them. But CVTs are not very common and many shops might not have a lot of experience with them. At the very least, however, I would call around a little. The problem is that you could take it to a dealer and there’s a good chance they they won’t even have anyone who knows much about them. They’re likely to tell you that you just need a new transmission. Then they don’t need to know anything except how to remove and old one and stick on a new one.