I think a lot of it seems more daunting than it really is. I'll try my best to answer some of your questions.
First off, Clockworkmod Recovery is considered a custom recovery because it takes the source code for the original Android recovery (something included with every Android device) and adds more functionality to it. The reason why it's necessary is because we can't flash 3rd-party packages with the original recovery. Furthermore, it allows us to really expand customization while simultaneously providing us a way to go backwards if need be. (Which brings me to the next point...)
A Nandroid backup, something done within CWM, is your safety net. You need it in case you royally screw something up. It's also very handy because it takes a literal freeze-frame snapshot of your device at that very moment (call logs, SMS messages, themes, browser history, etc.). You'll want to make a nandroid backup as soon as you get CWM on your phone. It's something that you don't have to do, but you'll regret it if something goes south and you didn't. Nandroids are also important because you can flash whatever you want and if you don't like it, you can simply restore your backup.
Technically, rooting isn't necessary. Having root access allows you the ability to change system files. This is disabled by default because manufacturers don't want to give us access to certain areas of the phone. This is mostly for our own protection. However, careful users shouldn't have any issues after gaining root access.
Installing a custom recovery on our phones is different than most devices. We have access to where the recovery is without the need for superuser permissions. However, if you want access to /system area on the stock ROM (useful for changing apps, themes, etc.), then you will need to root your phone. Custom ROMs, such as CM7 or MIUI, come "pre-rooted." If you're running CM7 or MIUI, you won't need to run any extra programs to gain root access.
MIUI and CM7 come as a complete package. You won't need anything extra after flashing either ROM. However, if you want extra features, such as battery mods or an overclocked kernel (which I advise against), then you'll have to find an extra package that you flash for your respective ROM.
Here's an example of going from the stock ROM to MIUI:
1) Add CWM to your phone
2) Download the MIUI zip package
3) Put the zip on your phone, turn off the phone, and boot into recovery
4) Make a nandroid backup (just in case)
5) Wipe your data and format /system
6) Flash the MIUI zip package
7) Once it finishes, reboot
8) Enjoy