|I was wondering if I should have (or do i need) DHCP running on this server?|
You need to decide where to run DHCP from. You have two choices, the router or your DC. Just don't run DHCP on both or you'll have problems. Which you pick is up to you as there's no real difference as long as the DHCP is configured correctly.
Does the subnet need to be a different network, like if the current router private subnet is like 192.168.x.x should I be using 10.0.x.x for the new private Intranet? Is that necessary or just pointless?
To put it simply, the answer is "no". I would just use the default class C IP scheme that the router has.
And is that possible without an extra switching device or do I need another router/switch??
Most SOHO Routers (Small Office/Home Office) have only 4 LAN ports. So, if your network is going to have more than 4 clients/servers/network printers you will need a switch. Figure out how many connections you're going to have and buy a switch with enough ports to cover that many, and some extra ports for expansion and growth. So, if you have 12 client/server/printes, buy a 16 or 24 port unmanaged switch.
If you're going to be running just one subnet, another router isn't needed.