Great idea! Here are your answers:
1. The cost of a 2TB SSD will be prohibitive - dont even know if they make them in such large capacities. Go with a SATA external storage disk. 2TB is fine.
2. Yes, the external disk must have its own little power supply - small plug-in transformer.
3. Hard disks are quite reliable these days. A 7200 RPM drive will be just fine.
4. 32 MB buffer should be ok - already mentioned 7200 RPM
There are two ways to share a hard drive among many computers:
1) Connect the drive to one of the computers and "share" it. If you do this, the computer that the drive is connected to must always be turned on.
2) Connect the drive to your network (i.e., to your router as you described). In this case, the drive itself (and the router) must always be turned on but there's nothing else that always has to be on.
If you want the latter, then do not buy any old drive in an external drive case. Most of these are designed to connect directly to a computer using USB, eSATA or FireWire. What you want is technically called a "NAS" device (Network Attached Storage) although not all consumer NAS devices actually use "NAS" in their name. Just make sure that what you buy is able to connect directly to the network.