TS runs on every 2003 server by default, and having all your Client Access Licensing information tied to a server that doesn't have the application on it is going to be more work than it's really worth. You'll really need to have that on the server that is running your application.
The choice is ultimately up to you about how you want to approach this. If you are just fine with managing access the way you have been, you don't have to get Active Directory, and you can run the Application on one server and the SQL database on the other (determining which one to use for which is another subject altogether),
If you *do* choose to install AD somewhere, you should run DHCP from the AD Domain Controller. That's just the best practice. If you don't, running DHCP from the router is perfectly acceptable.
Basically, it's going to be a balancing act between what you have to work with and what you want to achieve.
If you want the best performance you can get with what you have (which seems to be your purpose for getting the new server), figure out whether it's the application, the SQL server, or the number of remote connections to the server that is causing the slowdown. Once you have that information, you can determine the best way to move forward. If the application and terminal connections are causing things to be slow, move both to the new server and leave the Database on the old one. If the SQL database is using up more horsepower, move it to the new server and keep the application on the old server.
If you want, you could just keep doing things the way you are, move everything to the new server and leave the old one aside. That would be the one that would require the least work, but of course it also provides the least benefit.
