I'm going to visit Schengen area, Norway. Can I work remotely for my company back in India from there? Are there any restrictions on that?
The Schengen visa in and of itself doesn't give you any permission to work and there are indeed restrictions on that. If you were to come to the Schengen area to work for a brief period of time, you would need permission (implicit or explicit) on top of your Schengen visa. And if you were to apply for a Schengen visa specifically to come and work in the area, you would need to submit evidence that you do have the right to carry out this work with your application. Of course, those rules were defined with onsite work in mind but there is no blanket exception for remote work. That part is easy and unambiguous.
To muddy the waters somewhat, some work activities (checking emails, attending the odd online meeting) might be considered “business”. It's also likely remote work is not a focus for enforcement or perhaps even falls under some local exemption. It remains the case that, in general, (remote) work is not allowed under the Schengen regime.
No, a C-Visa ('short stay visa', 'Schengen Visa') does not generally allow any 3rd country nationals to work during their stay.
For 3rd country nationals, permission to work is solely within the competence of each individual member state.
For this a D-Visa ('long-stay visa', 'National Visa'), must be issued under which a member state can determine under what conditions a 3rd country national may stay (and do) within their territory.
One member state cannot grant permission for a 3rd country national to work in another member state.
Outside their territory, the D-Visa will be treated as a C-Visa within the Schengen Area. This also applies to a residence permit issued by a Schengen member state.