Actually, the improved video output happened a few years before the C64C. You can find "breadboxes" with both 5-pin and 8-pin DIN connectors.
Yes, all peripherals and software should work: disk drives, cartridges, keyboard mechanism (!) and so on. However, the main boards have been redesigned in terms of cost reduction, possibly soldering chips directly to the board instead of using sockets. Some chips such as the 8580 SID will run at 9V as opposed to the original 6581 SID which is a 12V device. You can search the Internet for a more detailed description, but internally you shouldn't swap chips without knowing what you do. Installing a 8580 in a breadbox may yield unwanted results if it is driven with 3V more than it is built for.
I'm not sure if the C64C runs cooler than the original model, but obviously it has a slightly different form factor.