Yes, NMS-BASE-250 can be used to install and run Extreme Management Center. Depending on when the entitlement was redeemed you may need to upgrade it through the portal to use it with version 7 or 8, but it will allow you in run Versions 6, 7 or 8.
Yes, the NMS-BASE-250 license allows you to install Extreme Management Center and manage up to 250 devices (managed IP's). I would, however, recommend installing the latest version of EMC, 8.0. There are quite a few improvements over version 6.3.