Does the Time Machine back ALL of my data onto the hard drive?

Time Machine is not archival. It is only ever guaranteed to be a copy of the current state of your Mac. While it will keep items in the older backups until the drive is full, it will start deleting things that are not currently on your Mac. You have no way of knowing when it will delete those items.

As long as you don't delete everything and back up to Time Machine, you can erase your drive and then restore from that backup.

Why do you think you need to "completely reset your Mac?"

If iCloud Storage is full, open iCloud Drive on your Mac and start deleting files from iCloud, or purchase more storage.

If you have Desktop & Documents in iCloud Drive turned on, then if you delete files from the Desktop & Documents folders in iCloud Drive, they will be deleted from your Mac. That service synchronizes your Desktop and Documents folders to iCloud Drive. Those folders are not separate copies of the same things on your Mac.

If you are backing up iPads and iPhones to iCloud, then those can take up a good part of the 5GB of free storage.

I was having a hard time deleting things because everything was stuff that I wanted. So I bought an external hard drive and backed it up.

You should move data that you want to keep but not store on your Mac to an external drive. You should not use Time Machine for that purpose as I noted above. Just copy your files to the drive and then back up that drive along with your Mac. If it is connected to the Mac, Time Machine will back it up along with the Mac. However, that is a bit more difficult to do with a laptop, just as backing up is difficult on a laptop. You've got to periodically connect both drives and let Time Machine (or whatever backup you choose) back up both drives.

Does the time machine back ALL of my data onto the hard drive (that would be ideal)

Basically yes. Although technically it ignores a few bits of data (such as Cache folders) as they are not needed.

If the above is true, is it safe to delete everything? I know it isn't advised but can I?

If you mean delete from your main disk, the answer is this:

Yes you can delete files from your Mac and Time Machine will not remove those files from the backup (although it will note the fact that the files were deleted. But Time Machine will assume you do not need to retain those files as you chose to delete them; so when Time Machine runs out of space, it will eventually start removing those deleted file.

And in any event, if the files are only stored on the Time Machine drive, and that drive fails, you will lose your files.

So don't rely on it!

. and if I could, how do I? Should I just completely reset my mac?

You need to consider what you're trying to achieve. In generally the answer is that you need to delete files you don't need, or install a larger hard disc, or sign up to things like Dropbox that allows you to store files on the Dropbox server and remove them from your Mac; you can then recover files one-by-one as you need them.

. Also, my icloud storage is full, how do I clean that up? I couldn't find a way to do it on my mac so if there is another way (or a way to do it on the mac) please do tell me!

Pay Apple for more storage! Here in the UK it's free for 5GB and only 79p per month for 50GB.