I don't believe you've provided enough information for us to help you.
A couple facts and some questions:
The 2600 router won't pass DHCP requests by default. It needs to have "ip helper" enabled on the ethernet port pointing towards a DHCP server. An exception is if 2600 is setup for layer 2 bridging (you'll see bridge-group commands on the interface in question) --- then all layer 2 broadcasts (including DHCP requests) will get passed over that link. The reason for this is because routers block broadcasts by design. They connect multiple broadcast domains.
The switch will pass all broadcasts, and all ARPs for unknown PCs, devices, etc. out all of the ports until the device in question replies.
Spanning tree is likely a complete non-issue here.
The "uplink ports" on switches are generally "special." Sometimes they have a pushbutton switch (I think maybe the 2900 does), which allows you to auto-cross the ports so you don't have to use a crossover cable. In your case, both the PC and the Router are "end devices" and connect with a straight cable. There's nothing special about port 24 here.
You cannot have more than one DHCP server offering different network addresses in the same broadcast domain. If you want two servers serving up addresses from the same network range, you have to configure two separate address pools. (DHCP Handbook, second edition, by Droms and Lemon) There are also DHCP failover options for redundant servers.
If you do end up configuring DHCP on your router, make sure that the addresses you pass out are in the same network as the ethernet port on the router. Example: Router IP is 192.168.1.1/24. You might want to pass out 192.168.1.2 -> 192.168.1.100. The default gateway for the PCs will be the routers IP address and has to be DIRECTLY reachable. If you change anything on the router, make sure that some routing protocol like OSPF is configured to announce this change of network range to the rest of the network.
Is your router configured to route IP or bridge? Once again, look the presence of bridging commands like "bridge 1 protocol ieee" or "no ip-routing" and so on.
If your router is routing IP, then, is it configured for ip helper as described above?
If the router goes off to some corporate network someplace, please call IT immediately, tell them what you want to do, and they should be able to help you. Rearranging stuff like this can cause you lose connectivity, which would probably be a truck roll or at least multiple phone calls and troubleshooting.