Is it important that I connect my iPhone to iTunes? Why do I need to register my iPhone through iTunes? A pop-up shows up. What are the things that the iTunes can do that the iCloud cannot do? Can I restore to an older firmware using iCloud? For example I downloaded an older firmware, because I want to downgrade, the iTunes has an option where you can choose the older firmware you downoaded then restore to it, can iCloud do this?
If you want a full local backup (including music and everything), you would wanna use iTunes unless you have iTunes Match and icloud backup. Restore from icloud is painfully slow if you have a lot of stuff. The registration on iTunes is just to link the phone to your Apple ID. No you cannot install a previous version of ios via icloud. For that matter, depends on how old the firmware is, you might not be able to install it from iTunes either.
I saw a video where he downgraded his 5s from 7.1.1 to 7.1 using a downloaded firmware from the internet and restoring it using iTunes by choosing the older firmware that he downloaded. Is this possible for the iCloud?
icloud is not an online version of iTunes, the purpose of icloud is to backup and restore contents, so no, it's not possible. All the firmware upgrade/reinstall must be done thru OTA or iTunes.