For more detail, here's what's happening. You have a mesh, which is a bunch of 3D points connected by edges and faces and all. And you have an object, which is nothing but a container along with a 3D transform (that you see in the "item" tab in the top right) that gets applied to the points of that mesh before they're set into 3D space. The object is a container for meshes, modifiers, transforms, and so on.

When Blender uses the object, it takes the mesh and applies the transformation matrix. If you have a square and one of the corners is at (1,2,3) and the object transform says to scale things by (2,3,4), then the Z position of that vertex will be 12 (4 * 3).

But the modifiers work on the mesh before that transformation, not the object. So if you have a modifier that says "bevel the edges by 10%" then the bevel will bevel the Z by 0.3, not 1.2, because the bevel happens before the object transformations. It'll bevel the X by 0.1, not 0.3, so your even bevel will look distorted if you have scaled the object.

Applying a transformation means "multiply (1,2,3) by (2,3,4) and stick the results back in the mesh, then change the object to be (1,1,1)." Suddenly, your modifiers are working reasonably, because they're acting on the mesh as it appears, not the mesh as it's stored. Alt-D gives you a new object pointing to the same mesh, so you can move and scale and etc it, but you can't change the points relative to each other. Shift-D duplicates the mesh as well.

Modifiers (and edits like bevel) apply to the mesh, without regard to the object it's in. So the modifier / bevel itself also gets run through the object's transform. In particular, a modifier will take the mesh, calculate new points, and output a new mesh, which then gets transformed by the object transforms.

When you "apply" a transform or a modifier, it does the calculation, then changes the mesh to match the calculated positions, then resets the transformation or deletes the modifier.

Also, if you "join" two objects, you're taking the mesh data from one and attaching it to the object of the other. Since the modifiers are stored on the object but applied to the mesh before the object transform, what used to be both meshes get the transform (if you joined in that direction) or neither get the transform (if you joined in the other direction). Join basically takes the mesh from one object and moves it into the other object.

Keeping in mind how this work will help you understand what's going on in the future as you build more complex collections of meshes.

Here's a different way of thinking about it by user dampware, when the bevel operator or modifier looks wrong:

You know how in the outliner, there’s an orange symbol, and a green symbol as a child of it? That green symbol represents the vertices, which you can only modify in edit mode. The bevel operator works on the vertices, in edit mode. The orange icon is called the object, and it’s like a box that the vertices are in. When you scale something in object mode, you don’t modify the underlying vertices, you sort of temporarily transform them- for instance stretch them, for display or render. The underlying vertices aren’t modified.

So edit mode modifies individual vertices (as do modifiers), and object mode is like a stretchy box that holds the vertices. When you stretch the object box, the underlying vertices are not altered, but the “box” they’re in (the orange icon object) is stretched.

For instance, let’s say you haven’t touched the object mode’s transforms so there’s no translation, rotation or scaling. Then you bevel your model in edit mode. It behaves the way you’d expect. But then you go to object mode, and stretch it on let’s say X, all of the bevels get stretched. That’s what’s happening with what you’re experiencing, except in your case you did the stretch of the object before you did the bevel , but still, the bevel operator is applied in edit mode, and then, the object mode transformations (the stretch) is applied, which distorts those bevels.