Ok, I have the following questions (for now):
1. Regarding the Intel NUC units. First off...wow these are cool. Second, so I need to add RAM and hard drive, correct? Will that go inside the NUC box, or will in be external?
Generally, yes you'll need to supply the RAM, storage (SSD is the way to go with these, at least for the drive that the operating system is installed on), and an operating system. Those all go inside the NUC box. I don't think any of the NUCs will fit an actual hard drive (meaning the bigger drives that have actual moving parts) inside.
2. Can I use an NUC for his primary PC and run the network off of it? You plug in Monitor, it looks like it could Bluetooth a mouse, keyboard, and printer
Sure. You'll likely need more storage than just the SSD in the NUC though. External hard drive(s) can solve that problem, depending on how much storage you need. (A very high quality 4 TB HDD can be had for under $150 these days. That's a lot of storage.) You can get a good drive like a WD Red, a $20 enclosure, and you've got a very capable external hard drive with a boatload of storage space.
3. On the Synology NAS units....it seems to me that a two bay unit with 4 TB drives would cover us for a LOOONG time. (don't have anywhere near a single TB filled yet)
That's probably true. For those of us who remember the days before PCs became common, it's incredible how much storage hard drives have now. Anyway, the reason to get something like a Synology NAS is not necessarily the storage capacity but what the machine can do with it. I haven't used them but I understand they're very handy.
I do think that backup should be part of your planning from the start. It should go without saying around here how important backing up data can be. An external drive dedicated solely to backup of your main drive is a good start.
4. Let's talk about PLEX baby... (lol)....but really, PLEX allows you to...what? Access your "stuff" from any device in the world? I looked at their site and step 2 is "add your media"...so do you have to load everything onto PLEX's system somehow? Does PLEX not access my NAS? PLEX seems promising.
I think Plex is installed on the NAS, and it lets the NAS machine handle things like transcoding of video/audio to then be streamed to other devices. I haven't used Plex, though, so hopefully someone can give you more information about that.