Well, it's more of a quirk about me. Most Bairs kills, but Shulk's range for it is insane... If he can land it, it's going to hurt.
The Back Slash thing I found out after a Shulk used Back Slash instead of Air Slash to recover which lead me getting meteored, but Falco's second jump and Fire Bird allowed me to get back on stage. Ike's Aether, on the other hand was weird. I think the guy recovered, grabbed the ledge, and the second hit registered leading me to get meteored.
Anyway, Shulk's Uair is a good anti-air tool due because it's fairly fast and because of its vertical range. Like most Uairs, he can juggle with it at lower percents and because of its range. I don't really know what Ftilt is used for aside from spacing and killing at higher percents. Same thing with Dtilt except I don't know when it kills.
Down Smash is a good punish tool because it lasts so long, but it gets easily punished too because of its start up and end lag. Up and Side Smash seem like read and punish moves. Hmm, all his Smashes are good for roll, spot-dodge, and maybe air dodge punishes.
Uair, Bair, and Dair are all commit moves as far as I know and Shulks might not use them unless they're going for the kill.
Vision is weird in a sense you can nick it and it'll send Shulk somewhere else or if it hits, it'll do fair damage or kill. Air Slash is like ZSS's Boost Kick.
Most Shulks I've fought start with Buster or Jump, but I've heard good that Speed is a good option as well. A Shulk will either switch to Jump or Smash for kills depending on if they want an easier time killing in the air and off-stage or if they want to kill on-stage and reliably.
I don't know, I think Shulk's a versatile character, but like pretty much half or two thirds of the cast, there aren't a lot of Shulk players that I know of.
I figured i could add a little input on Shulk given i main him. Typically, Speed and Buster are the artes a Shulk player will start off with in the beginning of a fight, i typically favor Speed and follow up into Buster normally, as Pivot Grabs and Short Hop N-Air are easy enough approach options to use to rack up damage from 0-40% usually, then i follow up into Buster and rack up the damage from there.
One of the main reasons that Shulk is probably high mid tier at best/low mid tier at worst is that despite Shulk's versatility, he requires alot of commitment and more importantly, alot of his moves are pretty laggy and punishable, unless you manage to space them out correctly, and even then, he's limited at times due to this. The only Aerials i tend to approach with are N-Air and F-Air, and N-Airs his only real safe aerial in a sense due to its low landing lag when not worrying too heavily on spacing. As you mention, his off stage game is pretty good, but hard to pull off if the Opponent knows how to dodge and avoid it.
His Counter is really powerful, but consider how Shulk moves while performing it and how its not neccessarily instant (the only instant counter that isn't a custom is his Forward variation, and that only works on the ground), it's decently hard to land if you aren't countering an attack on the ground. Forward vision helps in hitting an attacker coming in from the air if they don't immediately bounce off after hitting you, but that's about as good as his Counter gets with non grounded attacks, its about 50/50 if you'll counter in the air depending on spacing.
Back Slash isn't that good of a move, honestly, the only times i land it are if i use the initial jump of the attack to jump over a projectile and hit my attacker, or if i read them and land them. Typically, most of my Back Slashes end up hitting the back of the opponent, and that's basically from a read that i've found alot of people easily fall for. Basically, i'll grab an opponent, do a back throw, then Jump into the direction opposite from where i threw said opponent. Most of the time, if the Opponent doesn't hit the ground and tech, they'll jump back in the same direction towards me. If i read this, i can easily just reverse Shulk's momentum with Back Slash and land a hit on their back. I don't think it can be air dodged through it, as unlike most of Shulk's aerials, Back Slash's hitbox animation is long lasting, basically staying out in your entire duration in the air until you hit ground or die.
As for Air Slash, i haven't been able to hit with it often, the only time i can think to use it is if i perfect Shield and use it as a punish like an OOS option (except i let go of my shield rather than use it while holding shield button). It has surprisngly good KO power vertically, so im trying to use it more often.
Out of the MU's of the tops that i've played with Shulk, the one that i think he has the easiest time with (at least, compared to all of the tops) is probably Sheik, maybe Rosalina (havent played that MU enough to know if its easier than Sheik). Yes, Sheik can combo him like she can most of the cast, but with Monado Speed, Shulk can match her speed movement and improve his neutral game so that he has an easier time dealing with her. Not to mention, Monado Jump is the perfect tool for getting out of her edge guard traps most of the time, and Buster works really well on her. Yes, Sheik obviously still has the advantage, but Shulk has good tools to combat her, especially considering how he stands in the roster with his laggy attacks and commitment.
As for DIddy...You can probably imagine how he fares in that match up. :\