Looking for software that would enable me to cast a screen over LAN
I know of two bits of software that can do what you want.
They do what you are looking for, display stuff on their screen and do demonstrations on their screen. They also lets you spy on your pupils, blank screens, lock screens (useful if they are not paying attention), and shutdown computers at end of day.
AB Tutor
I have used this software, in the schools I have worked in.
On the negative side it is proprietary, and expensive.
Veyon
I have not used it. But it is Free Software and cheep.
See http://veyon.io/