Wow… I’m not sure where to start here…
Yes unix is multi tasking. User's can log into one box and using a variety of tool/apps work or store their information on another box. The name on the box is most likely a “hostname”. So to move (not physically) to another box you could "telnet" to box 3 and use your login in there (assuming you have a login there) even though you’re still at box 1. Maybe someone can suggest a beginning UNIX book or just do some google searches on client/server definitions and relationships.

For your second question, the admin there most likely has your login set up to point to a common storage space so any machine can access files. It makes it easier to manage and more efficient. If you think of one box as a single stand alone entity you’re missing the whole networking picture.
hth's