Ah I absolutely HATE ROB. I've fought a really good ROB pretty often.
Stages:
I wouldn't pick Jungle Japes, because he can camp on one of the side platforms and just spam the laser and his gyro. I couldn't do anything against that. Well the first thing is that you can never charge your giant punch and the second is that you have to approach him, which is nearly impossible. With his nair invincibility frames and his sidedodge + dsmash he's nearly impossible to hit.
Brinstar: Yeah, the best stage vs everyone imo. If I lose a match, I ALWAYS pick Brinstar as my CP and choose DK, and I never lost there. ROB can't camp there really good, because the stage is so small. Your smashes have great range and the hitbox stays out longer if you hit some of the stagethings (I love that). Bair is pretty cool and if you both get hit from the lava and you are near him upair is wonderful.
I don't know what else I could say about it... You can kill there at really low percents, but just look in Ripples thread about Brinstar to know more.
Now about the general ROB things...
If his gyro is on the ground you can attack him and it makes your hitbox stay out a bit longer (this once won me a match! He wanted to punish my fsmash lag and ran to me, but my smash lasted longer because I hit his gyro and he died, it was awesome).
Bair is really cool in this match, and you should know how to respond to sidestep + dsmash. Maybe downB would be good. But I mostly just shield until his dsmash is finished and punish him with a bair.
I love to spike everyone. And it's somewhat easy to spike ROB, because he can't air dodge during his upB, but he can always come back...
I think I use dsmash in this match rarely, which makes it a good kill move at high %, but I usually just kill with DKs awesome fsmash, because it's range is just too good.
If you want ROB away from you tilt spam is pretty useful, but his ftilt has about the same range as DKs, and is REALLY fast, so you can't ftilt him all the time. But I think his ftilt has more ending lag.
If you want to hit ROB in the air be aware of his nair and bair. Just try to trick him to use it and hit him when the attack is finished (or be really fast and hit him before the attacks start).
I try to follow him off stage sometimes for some bair kills, but most of the time he comes from either very high (which he probably shouldn't), or from below which means you can't do anything >_>
I think ROB has the advantage here... but not a big one. Maybe 55-45...