What kind of Backup Drive should I get?

You are very thorough! I just hope ransomware never hits your network, because, by the end of the day, all connected drives' data could be encrypted.

Amazon has a lot more reviews for this particular series of NASes. You may want to read through those first. I'm generally wary of any product which is rated below 4 stars on Amazon - the average rating for all Amazon products is 4.4, and if you can remove paid reviews ("I received this product free in exchange for an honest review) it's still 4.2. So anything below 4.0 is far below average.

https://www.amazon.com/Cloud-Personal-Network-Attached-Storage/dp/B01C7JIO5Y

The better dedicated NASes are generally made by Synology and QNAP, and are sold diskless (you have to buy your own HDDs and insert them). All NASes are basically small computers (usually running Linux) which connect to your network and stay on 24/7. The cheaper ones tend to be slower, have less functionality, and sometimes are unreliable. As you go up in price, you get better (faster) hardware, more functionality and stability. Both Synology and QNAP have come up with stellar UIs which are relatively easy to use, yet still give you access to the full capability of the underlying Linux system should you so choose.

This site is the best I've found for reviews of different NASes.

https://www.smallnetbuilder.com/nas/nas-reviews

As noted above, one of the drawbacks of using always-accessible storage for backups is that malware can corrupt your backups. So I have a large external HDD enclosure (holds 4 drives, 12 TB at the moment) which I use to backup my NAS monthly. That is my offline copy in case my daily backups are somehow corrupted. This doesn't mean the NAS is useless for backups. Since all my computers automatically backup to the NAS, I only have to run a manual backup to the external enclosure once a month. Without the NAS, I'd have to plug my external enclosure into every computer every day to run the backups manually.

Since I use FreeNAS (software for turning a PC into a NAS - do this only if you need more processing power as it'll consume $20-$100 more electricity per year than a small NAS), I've played around with the underlying filesystem. Most of my files (media library, old stuff which will never change) I've made read-only so malware can't delete or encrypt it. You should be able to do the same with Synology and QNAP NASes if you're willing to tweak a little - create a new user - say mediauser. Copy the files to the NAS as mediauser, then set the file permissions so anyone can read the files but only mediauser can edit or delete them. Then never use the mediauser account unless you're uploading new media files. (They can also run a Plex server.)

Also, if the constant backup is possible how would it work exactly? Would it be creating just a normal backup file for each machine, or could I made it so that it essentially makes a disk image of each computer then updates whatever files are different when it scans for backup again?

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Use a backup program which allows (best) incremental or (second best) differential backups. You make one full backup at the start of the week or month. With differential backups, every day only changes relative to that full backup are backed up. With incremental backup, only changes relative to the previous day's backup are backed up. So the full backup takes several hours, but the daily backup typically only takes a few minutes and you can set it to run automatically in the background.

Unfortunately, most free backup software disables incremental or differential backups unless you pay for the software. Some of the ones which include it for free are Easeus ToDo, Paragon Backup, and Macrium Reflect, though I believe only Easeus gives you incremental backups for free. I've heard Aomei suggested as well, but have never used it.

The Macs come with a really nice backup program called Time Machine. It's basically incremental backups with a really nice UI for browsing previous backups (pretty much the perfect backup UI in my opinion). I've only used it with a direct-connected drive. I'm not sure if it supports network drives.

Bonus: Would there be a way to back up an external HDD that I have plugged into my Smart TV the same way? Or would I just have to plug that in through USB and backup that way?

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Presumably the data on that HDD never changes? Just connect it to your PC, back it up, and plug it back into the TV. Or better yet, if the TV has a network connection, configure it to use the NAS as a media server. That way you only need to keep one copy of the movie files around on your NAS, and your TV, computers, tablets, and phones can all play the movie using that one copy.

And would I be able to backup my ps4 HDD the same way too? Or would I have to take the HDD out of the ps4 to back it up?

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Seeing as there's a feature request from a week ago asking for the ability to back up a PS4 to a NAS, I'd have to say no. You're gonna have to back it up to an external HDD. If you can get the NAS media server thing working with your TV, you can use that external drive to backup your PS4.

http://community.eu.playstation.com/t5/PS4/Feature-Request-PS4-Backup-to-NAS-option/m-p/25052578