Just thought I would add my thoughts as I have just had my whole house wired with CAT6, and it is one of the best decisions I have made.
First piece of advice as mentioned earlier in this thread is get 2-4 ports at each location as it isn't going to cost any more (other than cables) to have this done.
As an example I have 6 behind my Lounge TV so the following devices can connect into the network TV, PS3, HTPC, Raspberry Pi, Foxtel (Sky equivalent in Australia) and one spare. I then have 4 in my office, two in the master bedroom and two in the media room.
Like you, I have all of my networking equipment in a discrete spot; in my case in a ventilated cupboard above the fridge which is in the centre of the house, this includes a Cable Wireless AC Modem/Router, 24 Port TP Link Gigabit Switch, and a Synology 1515+ NAS.
I know in your OP you mentioned getting a server which would work well and you could remote into it from another PC, Laptop, Tablet, Smartphone which would eliminate the need for a Keyboard Monitor and Mouse under the stairs. If you haven't already I would recommend looking at a Synology NAS instead, they pretty much do exactly the same thing, incredibly easy to set up and the SHR Raid system is fantastic and will provide you with 1 or 2 disk protection should one of them fail. I am thrilled to bits with mine, I run a Plex server which handles all of my Films, Blu-Rays, TV shows and work very well with the Rasplex operating system on the Raspberry Pi. I have my Google Drive and One Drive cloud services installed on it too, along with a whole host of different functions and features. It has an App Store equivalent called Package Centre and you can download your own applications like iTunes Server. You can use it download torrents and it has full VPN support if that is of interest too. But I think the best thing about the Synology is that it is so easy to setup irrespective of your level of technical ability.
I hope this helps.