If 'the global net' is accepted... then 'the global network' ought also to be accepted
It is. Did you try it and have it rejected. Or did it say an alternative?
Is the word global really necessary? The Internet, by definition, is worldwide.
I don't think it would be wrong to use the adjective here. Besides, not all phrases in Duolingo have to make sense to demonstrate a certain sentence structure, or for users to learn new vocabulary ^.^
The "worldwide web" is also accepted as a correct translation. Did you use it and have it rejected?