Jura V tvir Mbenan'f crg n pregnva anzr, it has a briefcase/"eligible for cabinet position" icon on its card, but it is not actually possible to offer it a job from its visit-pet page.

Typo during One Bite's quest, rot13'd just in case :
"Vg qbrfa'g znggre. Lbh'ir tbg gur frafr bs vg abj. Lbh'ir fghqvrq fcnprnuvc qrfvta, lbh'ir ivrjrq pbhagyrff fpurzngvpf, lbh'ir yvirq va n fcnpr fgngvba. Lbh xabj jurer gur qbbef ner. Lbh qba'g arrq fvtug gb svaq lbhe jnl."
"fcnprnuvc" fubhyq or "fcnprfuvc"
Late and a really specific situation, but vs Mbenan'f zbgure'f fcvevg unf npphfrq ure bs zheqre, V guvax gung fubhyq pbzr hc va ure zneevntr rcvybthr jvgu Pnewre (gung vf, gur bar jurer fur yrneaf uvf gentvp onpxfgbel naq ersyrpgf ba ure rkcrevraprf vs fur rire xvyyrq nalbar). Vs fur nppvqragnyyl xvyyrq n snzvyl zrzore, gung'f eryrinag va gung fvghngvba!
We've been using this site to encode and decode spoilery text: https://rot13.com/
I can't help with all of these but I know some of them!
1. Stoyan is the best (Science 75). Lorkas is also good (60). Chapadoc and Robe of Ice & Starlight Needle would theoretically be good (60 and 50 iirc) but I don't know if they can actually be persuaded to take cabinet positions. Both Lorkas and Stoyan automatically give you multi-node travel.
2. I think Mirelle on Myrkka II is the easiest first impression you can get to fast—she tends to like you even if you can't pass the etiquette test in her conversation. (Sometimes I go straight from there to Mau before going back to Myrkka I, but AFAIK that's only a good idea if you want to make Erilotti Chancellor—she's good enough to give the extra social action, but she's very hard to impress otherwise. And you need either a couple sessions of Spirituality or to have hired Mirelle as Logothete of Law to talk to Muistis at all, with that approach. But Myrkka II - Mau - (Myrkka II again) - Myrkka I does give you a few more turns to build up skills to impress the Myrkka I characters.)
4. I think yes, Lukomir always announces on the same turn. (... Someone else please remember what turn it is, I'm only half sure I have the right number in mind.)
I think my favorite is the Muse's Robe, but I also like the Evening Gown, Golden Empress, Toolbelt, Fleet Uniform, and the Body Control outfit whose name I have forgotten. Not as fond of the Data Visor or the Holster Shorts, or the default. Haven't yet unlocked the outfits for Combat, Condition, Domination, Stealth, or Environment.
Only got the Data Visor once (and I was specifically studying to unlock it, wouldn't have trained that combination of skills otherwise). It was kind of a pain to unlock—I think I wound up spending 12 to 15 turns on getting it, as opposed to the six to nine for another outfit. Part of that was that I didn't think to wait until I had advisors for everything.
I found myself wishing the bonus was bigger for that one outfit, or that it unlocked at 10 rather than 15 of every knowledge stat, even though it's already significantly lower than the 25 threshold for the other skills.
I had a similar issue with an epilog screen with different text but the same illustration—wasn't sure if it was a big enough deal to be considered a bug, but it looked weird.
The full text in rot13:
Gur eriryngvba bs ure trargvp cneragntr envfrq dhrfgvbaf sbe Mbenan. Fur jnf envfrq Zlexxna, ohg ure oybbq jnf Abina. Fur unq pubfra gur Rzcver naq orra oyrffrq ol gur unaq bs gur Uvtu Cbagvsrk, ohg jnf gung gehyl gur snvgu fur jnagrq gb sbyybj?
Nsgre n crevbq bs fbhy-frnepuvat, Mbenan qrpvqrq gb nssvez ure pubvpr bs gur vqragvgl fur unq nyjnlf oryvrirq va. Gur arj vasbezngvba jnf na nqqvgvba, abg n punatr. Ure urneg orybatrq gb gur Rzcver.
Nf fbba nf fur jnf byq rabhtu, fur znqr ure qrqvpngvbaf ng gur grzcyr naq jnf znexrq jvgu gur genqvgvbany gnggbbf bs nqhygubbq. Gur pbafgryyngvba pubfra sbe ure jnf gur Tngr bs Urnira, gur cnffntr orgjrra ernyzf gung njnvgf nyy fbhyf va gur raq.
The second-to-last line (Urnira, gur cnffntr orgjrra ernyzf gung njnvgf) had its first and last letter just barely touch the frame, while in the last line (nyy fbhyf va gur raq.) the first word and the first letter of the second spilled over onto the curved corner of the frame.
Must-have: Interaction (especially Etiquette) and Emotion (especially Empathy—knowing who likes and dislikes me feels vital). Any time I skimp on these I suffer for it.
Useful: Piloting comes up more than I'd expected! Also Engineering and Arts, and Endurance specifically in the Body Control category.
I'm not yet at the point of writing anything off. I haven't yet focused on Stealth, but I've failed enough Stealth tests to see that it can be useful. Domination seems limited, but there are several quests where it also seems necessary. I haven't explored Combat, Condition, or Environment yet.
For the most part I neglect knowledge skills for Zorana and focus on hiring skilled advisors (with a couple of exceptions—I'll learn a bit of Spirituality just to make good impressions, I'm currently experimenting with Galactic Relations, and the super-Intrigue run I did was interesting even though it was a little immersion breaking that everyone had identical military/intrigue/spirituality stats). Science and Bureaucracy are the most vital in my experience—the movement speed and extra social action are both super useful.
Maybe if you could only check allies/enemies who you've met, using the system that checks for the visible opinion modifier? IIRC the stars/plusses/minuses only show up for allies/enemies that you've met... (Or maybe looking at unmet allies and enemies could use something like the mostly-white image that shows for Ethexae or Muistis before you've met them, and just have text that's "Name, Planet, Sector" with it?)
Your pet is important! Visiting them sets your mood to Bold (hawk), Indulgent (viper) or Negative (blob/maw of darkness). Since in between the social phase and selecting next turn's classes you have the mood picker, this actually means that:
Since winning people over is such a large part of the game, the Emotion and Interaction skills are very important. Emotion is boosted by Indulgent and Social, and Interaction is boosted by Compliant and Social. So I'd recommend the snake for a first run—it makes it much easier to build up those social skills!
You may be able to unlock a way to set your mood to Compliant in the same way as visiting a pet—you'll be able to reach Indulgent/Cautious the turn after, and Social, Negative, or Compliant again in two turns. If you manage to do this in a hawk pet run, you'll be able to reach any mood on the wheel in two turns, which gives you a lot of flexibility in learning.
Event: Agreed to Rippling Eternity's request during negotiations.
Problem: Rippling Eternity asked for a third elector for Heru Prime, the Heru homeworld. Rihlat, the tiny planet it represents itself, is not the subject of the promise (and if it was, it would be getting a second elector, not a third). (Also, during negotiation, Zorana's thoughts/knowledge refer to "the All-Seeing Eye", and I think usually the moon of Mau Prime is referred to as the Blind Eye? )

Ooh that is not who I thought did it in the original version of the scene!
Rot13:
Perrcl. (Mbenan'f qnq vf n ovg bs n fhcreivyynva, vfa'g ur?) V'q thrff ur'f "fgnegvat bire" jvgu nabgure pybar bs Zvynavn? Naq gur cynlre vf fgnegvat bire jvgu n arj tnzr, bs pbhefr.
Sbe ceriragvat vg: qbrf Mbenan arrq gb ernyvmr gung Qentbpra vf fgvyy nyvir?