Well, you're going about it the right way if you want to centralize all your data and share it among your LAN clients. This is it the only serious way of going about this as home networks become bigger with more clients. In answer to your questions;
1. A device like the Synology will walk you through this during setup, and make recommendations. If you've a fully loaded DS1513, it will default to a RAID5-like setup. With RAID5, all the disks are put into one large 'pool', with parity (essentially recovery) information written in stripes through all the disks. This parity information takes up space, so you end up losing the capacity of one of the disks in the RAID array (you'd end up with about 15TB formatted once RAID5 is up and running). With RAID5, you can lose an entire disk, and the system will keep running. You just replace the disk, and the system will rebuild the lost information from the parity data. Downsides of RAID5 are; losing the capacity of one disk, slow create/rebuild times, processing overhead to calculate parity information which means slower writes and generally RAID5 is looked at as overkill for home use, but, for simplicities sake for someone not sure on all this, it does make things simple.
2. NAS's do not have to run 24x7. Many bespoke NAS devices (Synology, Thecus, Drobo etc) have built in power saving modes, so the NAS itself will spin down the disks when not in use, and go to sleep if necessary. It will automatically wake-up when someone tries to access the data. These devices are low(ish) power anyway, and small, so can be tucked away. All management is done via a web console, so can be done from anywhere on your LAN. Generally, NAS devices of this type will have loads of other features as well, which you may, or may not, use.
3. You've found the biggest down-side, namely price. These NAS devices are (usually) very, very expensive, and yes, you do need to add the drives yourself, which in your case would push the price up to way over $1500 for the Synology. That's an awful lot of money, but if you want ease of use, discreet size and support, it's an option. Many here though build there own storage devices with different hardware, software and config. There's lots of choices, but you do need some knowledge to put all this together. It would generally work out cheaper as well, if you're prepared to put the work in.
Drobo's are meant to be pretty good, but I've never used one. I've used Synology's in the past, and they were ok, but I soon outgrew the one I had. Looking up the latest Drobo information, I think this would be a better choice than the Synology. The technology in a Drobo is very good, and pretty much everything is handled for you. Install the drives, switch it on, run through the wizard, and let it do its thing.