Is the size of a struct required to be an exact multiple of the alignment of that struct?

5.3.3/2

When applied to a class, the result [of sizeof] is the number of bytes in an object of that class, including any padding required for placing objects of that type in an array.

So yes, object size is a multiple of its alignment.

One definition of alignment size:

The alignment size of a struct is the offset from one element to the next element when you have an array of that struct.

By its nature, if you have an array of a struct with two elements, then both need to have aligned members, so that means that yes, the size has to be a multiple of the alignment. (I'm not sure if any standard explicitly enforce this, but because the size and alignment of a struct don't depend on whether the struct is alone or inside an array, the same rules apply to both, so it can't really be any other way.)

The standard says (section [dcl.array]:

An object of array type contains a contiguously allocated non-empty set of N subobjects of type T.

Therefore there is no padding between array elements.

Padding inside structures is not required by the standard, but the standard doesn't permit any other way of aligning array elements.