Align Auto

Hi Nazanin,

That is because of your Selection, before you Aligned.

You have Sync Selection on, which causes Face Corners, that share a Vertex, to all get Selected, when Selecting one of them:

You don't want that here. You want to disable that:

To be clear, what look like Vertices in the UV Editor are not Vertices, but Face Corners

Yes, as Martin says. Also I think you need to be on face selection when selecting a UV island to follow the active quads. 

I believe that has to do with the UV Smooth on the SubD modifier. Change it to wither one of the options and it should work:

Then you can just add an extra edge loop that holds the UV in place. Control + R on that inner part of the pot and make a holding edge there, that should avoid the UV's being drip down like that.

Indeed, as Omar said: