DecentM

This is still the case, as demonstrated in the video I just uploaded.
A workaround is to switch the triggered command block's type to impulse, and chain blocks react to them just as well, and it's only ran once.

I can confirm this, also happens when the selector is @a, optionally with a valid scoreboard value (like @a[score_test=1]).
While @p could produce an error, @a should work even with just one player.
Interestingly, /tellraw @p and @a works fine for me.

The video I just uploaded didn't want to start when clicked, so I re-encoded it, it can be played now.

Yes, I can confirm this is still an issue with 16w42a. I'll try updating the ticket in the future

Video from 16w42a

When hitting for the second time, the video doesn't show the button being pressed. That's because of it's framerate, it does get pressed like before, just quickly.

EDIT attachment

Can confirm that the dedicated server is having the issue too, and also that /gamerule doMobSpawning is causing it, however, gamemode didn't matter for me.

This still exists in 14w30c.

I did see that comment. I believe those holes would be loaded perfectly in the next few seconds. However, this world was loaded completely (I waied around a minute seconds while moving around), and there is water on the bottom, like it should be in an ocean biome.
Unless I'm being dumb, and not realizing that these are the same, I think this is a generation bug that happened in the internal server, and the other one is just how the server sends terrain to the client.

Both - It's constant.

Worldsave - Broken generation uploaded

I don't think this is a duplicate of said issue as the world was generated completely. I'll add the world save here.

My problem was not that only the half of the door got moved, because that's the intended command. Sorry if I wasn't clear, English is not my primary language.
What I meant to emphasize is that the upper part of the door snapped to the west side instead of staying where it was.

After some testing, I found out that it got fixed apparently. Now the top part just disappears as seen in the first 45 seconds (and how it usually works): http://youtu.be/YNvKPzqaU4U

I have just discovered this myself, and I wouldn't call it a bug.
It is probably not related to the issue mentioned, because we aren't talking about the position, but the way they render.
If I'm not mistaken, Minecraft looks at the texture it is given for clouds, an then renders each pixel from the png inside the world (each pixel representing one "piece" of cloud.
It is perfectly understandable why this is like that, and it was most likely intended. However, to the regular player, it is considered a bug, as noone would expect clouds to be smaller inside (reverse tardis perhaps?).
It would be a nice addition to the game if connecting cloud pixels were rendered as one, but this is outside of the bug tracker's purpose.

I also confirm this to happen with fill and setblock clocks.