About the vote in TPM, we didn't see the actual vote, we just saw Amidala calling for a vote, and the other Senators agreed. The vote itself took place when Amidala was on her way back to Naboo (she discusses the nominees for the position of Supreme Chancellor with Palpatine before she decides to leave, and congratulates Palpatine with his election after the battle of Naboo).

Concerning Qui-Gon: I thought Liam Neeson was one of the best things about TPM. As much as I would like it, I just don't think it's possible to put him in AOTC and ROTS. He could appear as a ghost in ROTS, but I think it would still feel tacked on. I'm all for making Qui-Gon's body disappear when he dies in Obi-Wan's arms. The explanation for why some Jedi disappear when they die and others don't, could be their state of mind. Qui-Gon, Obi-Wan, Yoda and even Anakin were at peace when they died. The Jedi dying AOTC or ROTS didn't die like that, they died violent and sudden deaths.