gridtools | Libraries and utilities to develop performance portable

I would go a) #2 and b) #1 choices.

If Map is a Singleton make it static.

a) #2 because a getter alludes inventory is a member, a function gives you more encapsulation and in future ability to tap into inheritance and interfaces should you extend the object model. Getter's suit single value much better than nested classes. Edit: I see you're accessing the getter via iGrid in #1, it's not very obvious to caller that's how you would set inventory. IMO it would be equally acceptable solution to use a getter that exposes iGrid if it was named InventoryGrid. If you have sub-nested classes this would be a better way to go.

b) #1 because logically to avoid duplication in a scenario where 2 grid classes need the same method a base/super class is one of the best ways. You might choose each grid to implement an interface if they have implementation specific CoordsToIndex functionality, if this is the case you could also use an abstract base method.

This function will be used by all Grids (InventoryGrid, TileGrid, etc...)

I think though that both grids will share the exact same implementation and one CoordsToIndex base class for both InventoryGrid and TileGrid is what you want right now (at this point in your design). I'm assuming that CoordsToIndex relies on data in both grids, otherwise your idea of decoupling and putting CoordsToIndex in the Map class is a good idea.

With Helper classes keep them more for app wide stuff: Config, File, Encryption, Logging/Tracing, extension methods, etc.

Your top 2 choices for both scenarios are good, so you're on the right track. One piece of advice is that interfaces allow you to use unit test mocking so you may want to write a few unit tests to confirm you can test your oop architecture.