Yes, for the indexer scenario dict[key] that is an exceptional failure (equivalent to indexing list[index] wher index >= length). The method to get the value of a key only if the key is in the dictionary is TryGet...

I think this design is the best one. It lets you have the program blow up quickly in all cases where the presence of the key is a requirement for the program to be in a valid state (which is common). Otherwise you'd be doing

   x = dict[key]
   if (x == null) 
     throw("Unexpected this is. And unfortunate");

Yes and probably make the most sense as .Net has no "undefined" value. If the dictionary has a the key with a null value it returns null, but no key throws.

With generic dictionaries (which everyone uses now days) you have TryGetValue which is a little weird since it uses a out parameter but pretty natural after c# 7 inline out params.

Checked access to c++ containers also throw ... https://en.cppreference.com/w/cpp/container/map/at