Makes no sense to me. Here is a summary of what I see from your ipconfigs.
The PC looks perfectly normal. It got an expected IP (192.168.0.102) from your router (192.168.0.1) and a lease of a week (a typical time period). If you hadn't told me differently, I'd confidently state that this computer is working perfectly fine on your network.
The laptop's ipconfig is totally confusing. The Dhcp Server and DNS Servers are correct (and match the PC) but the IP and Gateway addresses are public IPs, and the lease is for only 10 minutes. I can see how this computer can access the internet, but can't figure out how it got such a weird IP configuration.
Please make sure that your modem connects to the router's WAN (internet, modem) port and not one of the LAN ports. Incorrectly connecting to a LAN port is the only cause of this problem that I can think of other than a router that should be crushed and incinerated before it is discarded.
Second, if you are using any kind of turbo or speedbooster or whatever options on the router or laptop to get greater than 54Mbps, please see what happens if you turn it off because the Broadcom on the PC will not support it.
Third, see if there is a firmware update for the router (use an ethernet connection for updates!). If not, or if update doesn't fix it, reset the router to factory default settings and reconfigure it.
Fourth, look for driver updates for the wireless adapters.