1 It's a long story. Last candidates for Vermouth was Araide and Jodie, and Jodie helps Conan prove Araide didn't know detail in his father murder case, prove he is actually Vermouth in disguise
If you have time, read this
http://www.detectiveconanworld.com/wiki ... estigationEdit: If Conan believe Haibara, in the Busjacking case. It down Vermouth suspect to three people (Akai, Jodie, Araide). By "Vermouth" In Conan 's POV. It's only "BO Operative that came close to him" He did not know her name is Vermouth.
Later Conan conclude that in Pisco case, another BO operative was present because Conan rightly believe that someone give her handkerchief to Pisco. He able to deduct that said someone is Chris Vineyard and start to investigate her. Conan know Chris is a master of disguise (Sharon tell him in NewYork).
Last, after 4 Porsches case, Jodie and Araide bring an unconcious Haibara back to Agasa house. after they left, Conan search (off screen) and found listening device plant by Vermouth (Araide). This cut Akai from the list. Due to Mouri's investigation files were stolen. Conan again rightly conclude that Chris disguise as Araide and need th files to testify in court. But Conan did not trust Jodie either, so when she show up to pick Haibara up (before Araide) He has to ask her to prove Vermouth is Araide as I said in 1#
2 Vermouth contacts (send SMS to) the boss before he give the answer back. using the device in Conan's body, he will be able to determine what number and what content Vermouth send to the boss. Conan threaten Vermouth that, if she kill him or remove the device from his body, the information will be sent to his friend (Agasa) nearby, thus uncover the boss secret. Vermouth choose to put Conan to sleep and destroy that device, so she don't have to kill him and keep the org secret altogether.
If you have not watch beyond ep345. this should be a spoiler, but if you at around 420 then
3 Again, he hope when Vermouth get him (Which she get,eventually), she will contact the boss. He try to uncover the boss phone number by method described above. But Vermouth destroy the device so Conan got nothing.
4 Jodie make a cover story to lie her that it was only a kidnap case. Watch episode 346.
5
She impress him Conan impress Vermouth in some way, most importantly Ran and Shinichi save her from her death, when she disguise as a serial killer in the New-York case. Also Shinichi's word about there're no reason to not help someone.
6. (even)Vodka remember him. the point is he can't allow them to know he is alive.
7 I'm under the impression that "a secret makes a woman, woman" is right.