Not really clear what you want: Run minecraft on your laptop as a client, or use your laptop to host a minecraft server for 20 students each using their own laptop as a client. That makes a lot of difference.
What's the school doing to support you? If you running a server on your laptop, they most most likely will have to change their firewall configuration. If they don't no student can contact your server, and all your work and expensive are for nothing.
The best thing to do would be to ask them to host that Minecraft server on one of the school servers.
I'm going to help teach a class in a few weeks using minecraft. I was going to open a LAN server using my laptop. But the new update keeps crashing, I've been told my laptop was not meant for gaming (its a 4 year old compaq CQ57 it also runs pretty slow) so I think it's time to get a new laptop. But I don't know which ones are what I'm looking for, or will do what I need it to do. I need a laptop to run Minecraft with no lag, possibly run a server for about 20 students? But like I said I'm a T.A. in the class so I need it for my online class's and computer programs for school. Is there a laptop out there that can do all this under $400?