It's not clear what the photo is of. Is that a painted wall, and you spackled over the joint? What has siding and Tyvek got to do with it? You didn't fully explain what you're doing.
Based on only the photo I'm guessing you put joint compound over a joint between an old painted wall and new concrete board. That's not a good idea. You'll end up with a joint exposed that's hard to keep from cracking again, especially if it's not taped. If you want any hope of it not recracking you should use mesh tape and Durabond setting-type compound. But there's a much better way to do it: Clean the compound off. Bonding tiles to compound is just sticking them to chalk, essentially. It's much better to skip the compound and plan the last tile to slightly overlap the joint. The tile will then be stuck well to the concrete board and the unstable joint will be covered.
The stain is probably nothing to worry about, but it's hard to know for sure without knowing more. If you know you've stopped leaks then the stain will just be something leeching through from the wallboard.
Also, don't use compound or spackle in the joints of the concrete board. It defeats the purpose of the concrete board. If you think you need to fill the corners, do it with thinset and thick mesh tape.
Ignore Fred. He's just a mean-spirited person giving you a hard time about your use of English. He's not offering help.