> I really wonder what operating a huge company like that looked like in an era...
Lots of attempts to standardize procedures, etc., etc. - but East India agents far from home often had enormous latitude, and there were plenty of disasters and atrocities. (Not that either the British Government proper, or other European powers, were notably better. But they could certainly be worse - just look at the Spanish Conquest of the Americas, or the Belgian Congo.)
> ...the fastest way to transport messages was to have fresh horses pre-positioned ...
When there were enough short-but-important messages to be passed along a given route, they did have a far-faster-than-a-horse technology available - https://en.wikipedia.org/wiki/Optical_telegraph#India