Firstly with storage. I won't go into the DIY vs Off-The-Shelf NAS systems, probably a bit outside the scope of this post, however, since you asked, I'd lean towards Synology. By far the most important factor in storage though (whatever your option) is expandability. I cannot stress this enough, never, ever think that xTerabytes is enough. Buy or design around double or more the amount of drive bays you think you need right now, and expand later as required. It might get it a bit complicated if you want RAID 5/6, but at least you can deal with that without splurging out again on the NAS/PC cost. As far as the drives go, you need to lean towards the enterprise range of whatever brand you want. Don't be fooled into using desktop drives, or even bottom end "NAS" specific drives (like the WD Reds) they're cheapies - good luck if they last longer than 3 years. You're probably not going to get away with this cheaply (just on drive cost alone) otherwise it really won't last - or if you don't back that monstrosity up, you'll lose it too.
As far as the transport from NAS to TV goes, your wired ethernet/wifi will cover that, unless you mean hardware? DLNA might sound like a great idea, and the clients are built into every/most TVs, phones and tablets, but you're heavily restricted on file formats and codecs. I use either my laptop plugged into one TV, or a dedicated baby Atom box that is just a regular PC customised to behave like a media centre. This way, I get support for just about every audio and video format known to mankind, And I get Skype, and a REAL browser and just about any windows program out there, including my mapping software. Do NOT be fooled into the "my TV has a browser and skype too" argument - use them and see it's bollocks.
As per your question, you would need some sort of PC for each monitor. Be wary of the "transcoding" buzzword, most of what you think isn't really transcoding at all, it's merely "repackaging" (think DLNA transcoding). Real transcoding doesn't mean losing quality, and any PC wouldn't be fast enough to deal with it in real time anyway. Since you mention various formats, you automatically rule out DLNA too.
As far as multiple streams go, in your case, you're technically not streaming, but just reading the same stored video file from multiple clients. As long as your NAS, and network are fast enough to deal with that, it isn't going to be a problem.
As far as file/folder rights go, how you go about this would vary greatly depending on what DIY or off the shelf option you went for. Either way, both support login rights, where you restrict access read/write, read, or no access at all, down to username. I give read only access to the lounge room PC, because it gets mandhandled by various members of the family and other friends, and it's too easy for them to screw things up... The fact I use windows boxes to access the media makes this easy to sort out. Generic media boxes probably wouldn't work, have limited storage ability and DLNA isn't going to be suitable.
Lastly, and I found this to be by far the most important, you need a good, usable video/media menuing system. There are a few x86 "home cinema" menus available, (though I'm building my own). Having all your media listed A-Z is not an option, it's completely useless in fact. It's like having the world wide web with no search engine, just a list of alphabetically listed URLs. Billions of the things. You need to access the media easily, you need to have it sorted in a specific order (Such as TV series, or film sequels) you need to access more information if you choose, you need descriptions right there, you need search capabilities, you need to keep track of what you've seen or not. Some of us have 10,000+ video files in total, with no way of making any sense of it otherwise. Saving files on your NAS is REALLY easy compared to the database management, that by far takes the most work - because without it, you're sinking in a sea of files.