Ok I finally got the best ending, let me answer to anyone who wonders about the same question. (spoilers ahead)
To get the elcor scene with jack, you need to have the anal plug before you meet jack in the normandy as said in the walkthrought (before day 8), once you get the scene with the standard anal plug, you will get the possibility to buy the bigger one and have your way with jack (with sufficient influence and renegade points, read the walkthrought for more info).
ONCE THAT IS DONE do not forgot to look into the room of jack or the elcor, even if the game tells you there is nothing left to do on the ship go see there as you will find the bigger anal plug left on a table (go see at least once before going to sleep). That is when you will find jack riding the elcor, and the quest chain will begin.
I still have no idea if you need to have tali's influence to get the best ending but it didn't seem so (that said don't fool her with the suit, and a higher influence with her cannot hurt),
The companion where influence does not seem to matter at all are kasumi, EDI, samara, allers or Ashley. So better prioritize jack, aria, grunt (at least once to get traynor) on mission, and to accelerate things garrus or miranda (I would say they are secondary since you can always hang out with them at night and gain influence many times)
And I still have no Idea if the alliance mission gives you something or not in terms of credits.