the way i remember the story is that Duryodhana comes in first and sits by Krishna's head, while Arjuna comes in next and stands by his feet with folded hands. When Krishna wakes up, he sees Arjuna first (naturally) given he has just opened his eyes. Duryodhana remonstrates that since he was here first, he ought to be given the first choice. But Krishna mentions that he saw Arjuna first. He also bolsters his argument by stating that Arjuna is also younger and hence ought to be given first choice. Of course, when Arjuna hears the alternatives, he naturally plumps for having Krishna on his side, much to the delight of Duryodhana, who gets all Krishna's army.
I am also told that, he even vows that he will not lift any weapon during the war. During Kurukshetra he almost breaks his vow when he takes his Sudharshan Chakra by raising his finger to get rid of Bheeshma only to return back to normalcy. Can Mahabharata Ganabaadigal clarify on this!
Yes.. he does this when he sees Arjuna getting despondent on seeing Bheeshma's prowess, and questioning whether anyone can defeat him. To bring him to his senses, Krishna takes out his Chakra and says that he will kill Bheeshma. only for Arjuna to remind him of his vow and suggest that he himself (Arjuna that is) will do it.
Udanx, I am far from any ganapaadigal :)) But I kinda know this. Arjuna has a soft corner for his beloved grandfather and will not fight him with all his prowess despite the Gita upadesham. Bheeshma's might almost destroys the whole pandava army. Sri Krishna decides he has to do something to subdue Bheeshma and get Arjuna out of his affections, so he jumps down with his chakrayudha and rushes towards Bheesma. Bheesma immediately drops his arms and welcomes the Lord to kill him. Arjuna realizes he would be held responsible for Sri Krishna breaking his vow and takes up arms in earnest.
Arun, Duryodhona also had a daughter Lakna kumari - in her swayamvara Krishna's son Samba kidnaps her to marry her much to the ire of her father. Duryodhana refuses to officiate the marriage but fearing the ire of Balarama, his guru and Krishna's brother, finally agrees.
Both this story and the story of Duryodhana's son Lakna marrying Balarama's daughter with Ghatotkacha's help are folk tales, not part of the original Maha bharat. The Mahabharat only mentions that Duryodhana had a son Lakna who was killed by Abhimanyu in the war.