1) This is kind of true. RAID doesn't maintain file systems. It only maintains the way bits get written to the drives. The file system is further up the chain. But you're mostly correct in that if the controller itself died, you'd need a similar (but not necessarily identical) controller to rebuild the array.
2) Typically its just another HDD, of greater than or equal capacity, but I have seen instances where the controller was more picky. If you're running RAID1 on a 3-5 year old array and a disk dies, you are going to want to replace both disks anyway, simply because the odds of the second older disk failing are much much greater.
3) Get a NAS device. A ReadyNAS, a Drobo, something. Not for nothing, but if you are just getting into it and you need to ask these types of questions, you're better off sticking with something that's going to do the heavy lifting for you. Keep your focus on your work, not on the technology, leave that to the nerds like me and others.
4) See above, I have deployed ReadyNASes and I know many people who have and love Drobos. There are other options as well, I wouldn't be afraid of anything from QNap, Thecus, Buffalo, Synology, Promise, Lacie, or even Cisco/Linksys, Western Digital, or Seagate. They all have their merits and weaknesses, but all would most likely do the job you're looking to accomplish.
5) There are plenty. You just have to look around. Newegg, as much as I hate shilling for them, has a ridiculous selection, it can take a while to sort through.
Pick your capacity, pick how much growth you might need, and pick what interface you want, then start looking around. I'm a fan of ReadyNAS and Thecus over most others, but do some research and pick the one that's right for you. Its your data, do you really want to trust it to the opinions of some random nerds (or trolls) on the internet? :)
It is a standard feature in DSM.
You need the "Hyper Backup Vault" package on 1 of the Synology devices to act as the backup target.
On the other your run "Hyper Backup" to make the backups.
You can specify exactly what part of the NAS you want to backup and how often.
Alternatively (if you are comfortable with that) you can simply enable the rsync service on the target and do command-line backups with rsync.
In that regard DSM is just another Linux distribution.
Another option if you don't want to buy 2 Synology devices: If you have cloud-storage somewhere you can backup to that. For most cloud-storage system there is a sync app for DSM available.
Personally (3 Synologies at home) I combine both techniques: I backup the critical folders (mainly photos) of 2 of them to the 3rd Syno (using Hyper Backup) and from there I send an extra copy to OneDrive with the OneDrive cloud-sync app.
Best Answer
The Storage Manager will show the SMART status of each disk. Log into the web interface and go to Main Menu > Storage Manager > HDD Management.
You can also schedule a more in-depth SMART test using the Test Scheduler Option on this screen.