Quick summery: Otogi's dad got into a mini yami no game with Yuugi's grandfather over the Puzzle. Otogi's dad lost and his face was burned, this drove him rather insane and he trained Otogi for his revenge and set it up that Otogi would play DDD [manga name for DDM] against Yuugi for the puzzle. [making it look like Yuugi shoplifted so he was brought into the back room, stripped of the puzzle and had to play DDD against Otogi to get it back]

Yami Bakura shows up after the Puzzle's been shattered [Bakura had been worried that Yuugi had gotten separated from the group in the insanity of he Black Crown's opening and was all looking at the now emptied story in concern as Jounouchi was more or less dragging the group back to Yuugi's to look for him then. Short while later, Yami Bakura shows up, Ring pointing to the shop] and looks over the situation and tells Otogi that he's already lost and that he couldn't handle the Puzzle. [Yuugi is rather freaked out that Yami Bakura is there and Yami Bakura assures him that he's a friend/ally. [Nakama for the Japanese-language fiends. :)]] 

After Yuugi wins, Yami Bakura takes the Puzzle pieces back, tosses some insults/snirks at Otogi and hands the Puzzle parts to Yuugi. Insert sealing soul bit into puzzle piece here. :) He hands Yuugi the piece, then reverts to a very disoriented and confused Bakura-kun. Then Otogi's scary clown dad chokes Yuugi with the chain of the puzzle and drags him to a secret room and tries to put the Puzzle together himself. It makes him insane and he knocks over candles he had lit and here we have the fire scene.

Battle City:

The anime apparently decided that Bakura doesn't go to Yuugi's school anymore, or was on holiday or something because he's utterly cut out of Battle city until his yami is needed. Bakura's with the group from the very start of the arc, whenever the full group is there. He's also the reason that Anzu and co. are late for following Jounouchi and Yuugi through Battle City [he got caught up in his TRPG. *Grins*] So Anzu and co. didn't show up 'til after the Pandora duel. Bakura spots the Silent Doll as they're looking and is at first very 'Aa! Pantomime! Neat!' and then is kinda freaked out commenting how the guy is like a doll. Anzu drags him off then. They find Jouno soon after and start following him around. 

[Yami Bakura doesn't see the tablets at the museum, Isis doesn't lead him in to BC. Anime added that in as an 'Oh yeah, we need Yami Bakura for this later, almost forgot!']

When Malik arrives at Domino it's Bakura-kun who feels him coming, not his yami. Anzu calls for him not to get left behind and he scurries after the group again, and then has a conversation with Jounouchi about his occult deck. After Jounouchi follows he locator on his Duel Disc [handy thing, why'd they get rid of it for the anime?] Yami Bakura takes over and follows the Ring to where Malik is [pausing first to knock out someone and yoink their Duel Disk]. 

The convo between Yami Bakura and Malik happens outside the aquarium, there is no pissing contest, Malik just says 'work with me or die' more or less, I believe. Yami Bakura does the comment about having an easy host and breaking up Yuugi's friendships, then says his answer to Malik's 'work with me or die' is 'this': Pulling out a knife, licking it, and then stabbing himself in the arm with it, cackling like a maniac. He then retreats into the Ring with the comment to Malik that he can borrow his host, and Bakura collapses.

The section where Malik is carrying Bakura-kun is mostly the same, except that the kid that Yami Bakura stole the disc from is back with a friend and accuse Bakura of stealing it, Bakura [who I think it part his yami there judging by the 'aw crap' expression briefly] says that he doesn't remember how he got the disk, the kids go to beat him up, Malik steps in to protest that Bakura's injured and gets decked for it and then Jounouchi pounds them. Jiichan takes Bakura to the hospital.

Next: Vs Ghostboy and Battle Ship and beyond.