Unfortunately nope. What makes no sense is that he can breed with Hariyama but can't get Knock Off from him (even though Timburr can learn it from the B/W2 tutor) but Machamp can, also Conkeldurr and Hariyama can both breed with Hitmonchan but can't get Ice Punch from him but Machamp can.
It doesn't cost a lot to replace the batteries [there's like a five pack of the batteries on Amazon for five bucks, and a triwing screwdriver's only about a buck on its own; both seem to have free shipping], but I was mildly terrified of destroying my cart in the process ^_^
Your Pokemon Bank account is tied to your NNID, not the actual games. For example, I have a physical Pokemon Y and digital Pokemon X, and I can use Bank to transfer between them (though it doesn't count as a trade).
According to the Bulbapedia article, mothers or fathers both can pass down TM/HM moves, and mothers' moves take priority... but it seems exceptionally odd that they would cancel each other out if they both know the move.
New question #1: is there any way to get the PSS to stop nagging me to try new things? Especially when it thinks I need to try out Wonder Trade or the GTS... between the two I've used them just shy of 180 times [I keep every Wonder Trade or GTS 'mon I've gotten and I've nearly filled six boxes of the things]. It also keeps wanting me to fill out my profile survey, when I've answered every question. It's kind of obnoxious that it keeps nagging me like this and I would love for it to stop.
New question #2: is there some reason HM moves no longer seem to pass down to the child from the parent? I was breeding Hawluchas, and *both* Hawlucha parents know Fly. The children all lack Fly and instead have some of their level 1 moves instead. No idea why
