1. Stoyan is the best science advisor with 75 science, followed by Lorkas and Chapadoc with 60 each. All of those will instantly unlock enhanced movement for you. If you do an intrigue 100 run, you can see everybody's exact stats without having to actually offer them cabinet positions. Combine it with high media to easily see the stats for everyone at once (via the imperial poll feature). I made a spreadsheet to keep track of these and help pick my cabinet.
2. An important thing to keep in mind in this game is that people don't have to like you to vote for you. It's fine to make a bad first impression with someone if you're going to win them over via a quest or cabinet position anyway (the exception is if you lower their mood all the way into hostile, at which point they stop talking to you, but with reasonably balanced decisions most electors aren't going there off of a bad first impression). Also note that once you've been assigned an elector's quest, you can still complete it even if their opinion of you drops enough that they don't want to talk to you.
3. You definitely don't have to stay in the same star system until you're ready for the quest. I typically head to Myrkka I as soon as I can move and grab Ethexae's quest before she turns hostile, then immediately leave the system to go recruit my cabinet members. I come back and complete Ethexae's quest on turn 23, right before Agonaglia. At this point she may be hostile and unwilling to talk with me, but once the quest is complete, she switches to positive and pledges her vote.
4. Yes, Lukamir always shows up turn 10
5. This one I don't know, I've been too focused on getting the 29 vote run to investigate the secrets in the game lol