In the manga the time of his capture is unspecific, but it was either during the time skip or after Gaara's capture.
In the anime it was during the time skip.
Well, I don't recall that someone specifically mentioned in the anime when Han was captured, the only clues we have are Deidara's words to Naruto about the previous Jinchuriki Akatsuki captured, but these were in both manga and anime. What's more important, he didn't really specify if it was in part 1, during the time skip or lately.
It makes sense, but that would mean that they started to capture the tailed beasts at the beginning of part 1, since Gaara had to be first. It wouldn't be impossible,but i find it odd since between the introduction of Hidan and Kakuzu (when they captured the 2 tails) and Sasuke going after Bee, they had to capture 5 tailed beasts.
Also, they went after Fuu (who had the 7 tails) before part 2 took place, at the second chunin exams. But i don't know how canon that was since i only saw it in the anime. Was that novel based?
He was still there per the Akatsuki's orders, he still has to follow orders despite personal goals.
To my knowledge the manga never said that Pain tasked Itachi to capture the Nine-Tails then and there. The only source of information regarding the matter is from the anime. Pain intended to have someone go investigate Orochimaru as his whereabouts was unknown after killing the 3rd, so Itachi volunteered, to which Pain also tasked him to check on the status of Konoha's jinchuriki.