1) If I buy it can mutiple students use is on different computers or is it only restricted to one computer?
Cloud membership is for individuals. One person, max two computers.
What you have in mind sounds like Creative Cloud for Teams
http://www.adobe.com/products/creativecloud/teams.edu.html
http://www.adobe.com/products/creativecloud/faq.edu.html#ccm-teams
2) The work the students do, does it have to be saved on the "cloud"or can it be saved on the computer?
Work is always saved locally (same as traditional licensed products) but can be synced to the Cloud if you wish to use the Cloud storage
http://www.adobe.com/products/creativecloud/faq.edu.html
3) Lastly, when I buy it do the apps download on the computer or do I have to use it online
Cloud apps are the same as normal software purchased online - downloaded and installed locally
http://www.adobe.com/products/creativecloud/faq.edu.html
"Because your Creative Suite applications are installed directly on your computer, you will not need an ongoing Internet connection to use them on a daily basis. However, you will need to be online when you install and license your software, and at least once every 30 days thereafter. The software will alert you when you need to connect to the Internet for a license status check."