If the west proves itself incapable of doing anything about the Houthi problem and assuming there's a good enough incentive for Russia, there's nothing stopping Putin from using soft power to pressure Iran to get the Houthis to stop acting like niggers, all without firing a single shot.
Why bother chasing goat herders across the desert when you can simply negotiate with their owners? Again, assuming there's a good enough incentive to do so. As of now there's zero incentive. But once the US vacates their seat and pirates start ruling the seas again, expect a lot more calls to the Kremlin saying "Vova, we need help."
I doubt it, a massive buildup like that would require a degree of coordination and competence that is unheard of by Indian government standards. They'd be better off making security deals with other nations.
"Modernization" is a dirty word for the Indian military. Just like "standardization" and "indoor plumbing." Look up their military procurement process for a good laugh, they still can't decide if they're using NATO or commie gear.
India's actual contribution to maritime shipping is a lot more, uh, humble.