This is great that you would like to work in Teamwork with your students - I'm sure they will enjoy it and you can come up with great designs together!
About your questions, allow me to go through them one by one.
BIMcloud is the paid production plan of the BIMcloud family where all users must be able to reserve a so called BIMcloud User License. Same applies to BIMcloud as a Service, and the articles you pointed out are discussing options in BIMcloud as a Service. Graphisoft ID licensing is available only in this situation, on-premises BIMcloud does not have this option.
If you would like to use the basic functionality for free, you need to have BIMcloud Basic, where you don't need additional licenses, but a valid Archicad license (can be EDU license too).
2. IP address and networking
BIMcloud Basic can be used only via LAN (BIMcloud Basic and all clients (= all students and you) are on the same network). This can mean that you have a dedicated computer for BIMcloud Basic in the school and whenever students are in school, they can connect from school computers to BIMcloud Basic.
According to our EULA: "Special Conditions: BIMcloud Basic's usage is limited to local area networks (LAN / VPN). Using BIMcloud Basic over wide area networks (WAN / Internet) is strictly forbidden. BIMcloud Basic can be installed on on-premise hardware only. BIMcloud Basic is not allowed to be installed on / operated from cloud based virtual computers."
If you wish to access your BIMcloud Basic outside the network, you can use VPN, but for this you will need help from your IT I believe and it might be not that obvious. If you wish to do it on your own hardware, the whole situation is even harder, because you need to configure everything that way that it is available for your students via VPN.
I mentioned earlier that we have BIMcloud as a Service, where us, Graphisoft hosts the infrastructure and manages updates, all you need to care about is the user and project management.
This is another paid product plan option, which might be too expensive for you to pay. Since you are working in education, I suggest you to contact our Italian office and discuss the available options with them. I'm sure they will be happy to assist you and you can find the best option to serve your students need for the project.