Question

I would like to know if the ordering of the elements of a std::vector container is always preserved after erasing an arbitrary element (for example: push 0,1,2,3, erase 1, get 0,2,3).

Answer 1

Yes, it is. Also make note about the complexity (§ 23.3.6.5):

Complexity: The destructor of T is called the number of times equal to
the number of the elements erased, but the move assignment operator of
T is called the number of times equal to the number of elements in the
vector after the erased elements.

and iterators invalidation:

Effects: Invalidates iterators and references at or after the point of
the erase.

If you do not want to look through the standard to check such details and do not feel confident then have a look at this reference site.