Hi Andy,
Live Mesh will have the option to install custom applications directly in the mesh (you won't be able to install normal software like Office or Firefox) although this functionality is still in closed beta as part of the "Live Framework CTP" (check out
this page and others on that site).
As for the installed client following you around, unfortunately that is not currently possible. The next best thing (and nearly as functional) is the Live Desktop which can be accessed at
http://www.mesh.com which will allow you to download / upload files to your mesh and use remote desktop to access your mesh devices (some network restrictions may apply though, depending on your school's network).
Hope that helps,
Oren