Your last question first - yes. However, most of the CCNA is specific to the Cisco IOS. However, it does make you understand subnetting throughly which is important for networking. Most server admins don't really understand subnetting ... so if a network engineer gives them a /26 subnet mask they just kind of glaze over and most network engineers soon learn to give them the default gateway, network mask, and IP address and when it doesn't work to always check that first.
However, since you say you want to do "network admin work with servers and workstations" I would point you in the direction of learning and getting certified on Network Operating Systems instead of CCNA. What I mean is Windows 2000 MCSE, xNix certification on some kind of Unix OS, or Novell.
If you go the MCSE route, you can pick up Workstation certification along the way.
While CCNA will be helpful to you in getting a grasp on the lower layers of networking - routing, how TCP/IP works, subnetting, etc., it won't really help you very much in pursuing a Network Administrator position which deals more with Servers and Server Operating Systems.
Hope this helps.