Yea, duplicate IP addresses lead to some funny business. Toss a coin to see if a network packet will arrive basically.
The solution is to adjust the DHCP range or use static DHCP on the router. The latter just means that the router will assign the same IP to the specified computer every time.
That project is even a bit more involved because it looks to power an Apple display from a generic dvi video card. Going from DVI to ADC should be even simpler than the project you mentioned!