I've never had a lot of problems with assassins, except in one specific run where I took a decision that I knew was going to make a lot of people very, very angry at Zorana, and my general angle is generally Zorana Being Nice, so I don't know how much I can help, but:

- on the whole sticking to the left half of the mood wheel seemingly is less deadly

- Empathy/Etiquette can be enough to avoid assassination attempts on the left half, I definitely find Rhetoric very helpful when it comes to making a good impression.

- Physical skills - Condition and Body Control - I tend to consider as defensive skills against right half assassination attempts of the mood wheel. You usually need another skill (Layout, or Guns), but if they're sufficiently high you can make it through

- just reiterating what was said by previous people, visiting Korundi can put you in Cautious, which is generally but not always safer, and the snake puts you in the safer half of the mood wheel as well. 

- there's a few choices Zorana can make during the game that are offensive to some group or that some people will interpret as Zorana not caring for them. If Zorana shows lack of care for the same group a lot, then it'll be easier to pick up Electors from people who don't belong to that group, but that group might easily go from merely disliking Zorana to actively wanting her dead. Conversely, trying to balance out favor/disfavor means assassination attempts are less likely, but Electors might be less easy to pick up.

Personally I've found that what works best for me in the early game is getting a skilled cabinet from the Tharkan system and largely considering Myrkka II a wash. As far as I know you can't talk to the rival guy! He's not interested in being chummy with Zorana.  There's a trick to getting the admiral on board that I use a lot (you need a little Piloting and a lot of aplomb) which I find cost-effective but I don't know if it makes a lot of difference Elector-wise.

Do you do a lot of retrying from the start of the turn/the previous turn? I've found that a number of assassinations are easy to survive or avoid if you go back only or or two turns - if there's an assassination attempt in Compliant on turn 16, you could go to Social on turn 15 so on turn 16 you can only pick between Indulgent and Excited, and sometimes assassination attempts can be survived with an emergency double work session in a key skill. I did a lot of reloading in that playthrough I was facing assassins all the time.

Well, in order to explain what's going on, I'm going to have to talk about some things that you can never see "in game." Assassinations are never "pre-planned"; they happen because somebody really hates you (I've gone through several games with no assassination attempts). Initial introductions in every system are extremely important in making people like or dislike you, but that shouldn't be too hard to interpret. However, both during system introductions and during more traditional start/end of turn events, choices and skill checks can result in favor changes among 10 factions that you are never told about (there may also be racial-wide changes). Electors are affected by these factions, so when you make a dialogue choice that upsets the xenophobes, you're creating more enemies for yourself, and you don't even realize it.

This just to say that it's extremely hard to advise you, and I'm just going to put more pressure on you, because seemingly innocent dialogue choices may lead to your demise, because an unseen faction got upset. Generally speaking, if there are dialogue choices with skill checks that you can complete, you'll want to choose those over failed skill checks.

My initial winning strategy was to have a pet viper and focus on my social skills, interaction and emotion, with emphasis on etiquette, rhetoric, allure and menace. Since blackmail was nerfed, menace is probably not as good these days, but empathy is always useful to gauge affection, so that's a viable substitute. It sounds like you're on the right track doing something similar. Don't underestimate charming someone if you can pull the allure test off, it gives a juicy affection bonus to that elector.

One useful technique for avoiding assassinations is to take the snake as your pet so that you can set your mood to indulgent and then move it to social. I've never seen assassins on the "social" mood, and I did a run where I was specifically trying to bait assassination attempts (that Zorana survived 10+ assassinations and got 8 electors disqualified from voting lol). That said, it's possible to go through the whole game without facing a single assassination attempt if you do a great job at keeping everyone happy. Saving at least once a turn is definitely very helpful, though. When you fail a check the game tells you what you needed to pass it, and often if you go back a turn or 2 you can train those skills. 

Unless you're going for specific scenes that require a very high value in a certain stat, I've found that the knowledge skills aren't very helpful to train, it's easier to just get those stats from your cabinet. There's not really any single stat that's a "must-have", though, most checks can be passed in several different ways. There are a lot of checks for empathy, but most of them are just for telling whether or not someone likes you, not actually influencing people's opinions. Rhetoric and especially etiquette also have a lot of checks, but many of them have pretty low requirements. I've found that just grabbing student level (30) in both is generally enough to keep from offending everybody. 

Personally, I've never recruited the admiral via spaceships (yet). Having "survived an assassination attempt" (which you can get for free by having the Thresher sleep in your room), "played ritual Starfall" (which I already like to do to recruit Ethexae) and the wrestling accomplishment from either Lorkas' or Radomir's quest is enough ti impress him.

I don't know the exact mechanics behind assassins, but the thing I've found helpful in picking a cabinet is making sure that none of the members are enemies with each other, because that was explicitly said an update to be something that tanks their approval of you. Having a high skilled cabinet doesn't inherently increase assassination attempts, it's just that some of the electors with the highest stats also have a bunch of enemies. Also, if you haven't found out, getting intrigue to 100 once is pretty useful for figuring out exactly what each elector has to offer you.
 

Just to reiterate what Franka said: There are no fixed assassination attempts in the game. Every single time someone tries to kill you, it is because some Elector in the game's list of Electors has reached a point of disapproving of Zorana *so badly* that they outright want her dead. Almost every assassination attempt, if you survive it, will then reveal which Elector was behind this particular attempt (and then remove them from the game - this is one way to have fewer enemies voting!) So there's no way for us to know who was trying to kill you in your playthrough because it could have been any of them.

After this point I'm going to talk a little about game mechanics. Some of this qualifies as mild spoilers because people don't necessarily know this!

Not every Elector is capable of doing assassinations. For example, Ethexae, Muistis, and Lukomir will never try to kill you. This is partly because it would screw up the plot if we had to remove them from play, but you might also just assume it's beneath their dignity. They may or may not be the only ones. For the same reason, na Ryrpgbe pnaabg gel gb xvyy Mbenan vs gurl unira'g zrg lrg! Vg jbhyq oernx gur cybg.

Every Elector has a TYPE of assassinations that they are into. Certain Electors might only be interested in killing you via a stab in the back and others only want to shoot you in your face.  And those mood scenes open themselves up to specific types of assassinations, so if only one Elector wants you dead you can avoid that death by staying out of the mood scenes that involve that method. The reason people think that the left side of the mood wheel is safer is that while you can get assassinated on the left side of the wheel, the Electors who favor social assassins tend to start with higher approval scores of Zorana in the first place and are more easily pleased by the "generic nice" dialog choices that new players are likely to choose - options that will absolutely tick off other Electors. 

Just to emphasize that last bit - remember what kind of Empire this is! Remember what these people are like!  Being 'nice' will earn you enemies. Being noncommittal will earn you enemies. I've watched multiple people do playthroughs where they wondered why Yakau hated them by the time they met, when they'd been nothing but "nice" to everyone... THAT'S WHY!

You may find it helpful to do a run where you get max Media skills - at max level you can see everyone's approval ratings updating which will help you get a sense of how things are changing. Or maybe that's too much info and you'd be happier doing a run where you just go straight to Myrkka I and stay there until you've made friends with everybody (except Lukomir obviously). It might be less overwhelming to just focus on 'how can I win THESE votes' for a while.