How can I be sure that the two objects below have their respective bases at the exact height from the green (y) line or one another in blender?
In the upper right corner of the 3D viewport click Options then Transform only Origins.
Enable snapping and in the adjacent menu pick Face.
Rotate to a view on the underside of each object, select them in turn and [G]rab the origin and move it anywhere to a bottom face where it will rest because of the snapping.
Select the Item tab on the right side of the 3D viewport and change the Z Location for each object to zero. You do not need to deselect the Origins transform from the first step for this procedure as the Item values do not change the position of the origin in relation to the object, but rather the object as a whole including the origin, and this location points to the object origin.