thought i'd jump in here ... i just skimmed most of the thread so sorry if I miss something ...
I wouldn't disable DHCP completely. Most of these home broadband routers are set for a large static range on top of the DHCP range. For example, Linksys boxes are set for the following:
192.168.1.1 = Router
192.168.1.2 - .99 = Static Range
192.168.1.100 - .254 = DHCP Range
Now, as a general rule, any device that will be on your network all/most of the time would benefit from a static IP. I'd sugest assigning your Desktops and replays Static IPs. You can pick any IP in the static range. Check your router for this, I'm not sure how Belkins are set up.
This way, all the machines that are there are there all the time will always have the same address, but DHCP is still available for laptops, and devices that are brought in temporarily.
Technically, devices that are on all the time should always have the same address even with DHCP ... but it's better to set a static and know for sure than to rely on 'should'.
Regarding your PC in the kitchen, if it goes in to hibernate it certainly could change IPs. The way DHCP works, when your machine requests an address, it is assigned an address and a 'lease' time for how long it's allowed to use that address (usually about a day with home routers). When this lease time is halfway up, your PC will request another address. The DHCP server will USUALLY extend the lease for the existing address.
With that said ... your kitchen PC has an address, it goes to sleep. Some time goes by, it's lease expires, then it gets turned back on. The PC will send out another DHCP request to get another address. Now, if another device had come onto the network between the time the kitchen PC's lease expired and when it requested the new one ... its precious address would be no longer available.
In not so many words ... yes, your machine will request a new address when it comes out of hibernate/sleep ... and yes, it's very possible it could get a new address.
Again, my recommendation is assign a static IP to your desktop, replay, and any other device that is there all the time ... and leave DHCP for roaming devices.