All containers can (at least theoretically) support modifying the container while iterating. You just have to adjust the iterator to take account for the changed container. C++'s std::map::erase(iterator) returns a new iterator for exactly this purpose - the iterator pointing to the next element before the operation but one that is still valid after the operation. Unfortunately you can't use it with range-based for loops even though they still use iterators under the hood but c'est la vie.
Sure they can, but if you define an interface that allows this, every container type implementing it must support it, and it is probably gonna be rather slow operation for the effort. That's why the question is not "can you do this?" but "why would you want to do this?". I can't think of a good reason for generic containers, if you need something like that, it should be a purpose-built data structure that efficiently supports it.
The STL containers are full of "features" that containers in other languages just do not support, yet it's worse to use in my opinion.
