1) There are soft skills displayed in your presence that you are conscious of, and those working behind the scenes. Knowing how to manipulate others in Machiavellian fashion is also a soft skill. As has been discussed before, here and in other places, skill in deception is associated with leadership skills. So when one asks how important soft skills are to career advancement, I would rate the ones that others can see and be aware of a 7-8, and the ones operating behind the scenes a 9-10. Knowing who and how to schmooze is very important. Being your own lobbyist is a soft skill.
2) Any work that does not involve one person acting entirely alone will depend very heavily on the quality of relationships with others. Not just whether you "get along", but also on the ability of the co-workers to persuade each other, to foster trust in each other, to facilitate motivation in each other, to cooperate in persuading others, and to accurately and clearly communicate tasks/goals/strategies/methods/requirements to each other. Yes, there is some smarts required in there, but all of that rests on "soft skills". The sort of job performance one might present as meritting advancement will depend on soft skills.
3) Of course self-promotion is essential to career advancement. Keep in mind that those in pursuit of a "career" keep moving around, looking for that next promotion. While there will be rare instances where there is enough corporate memory to keep track of your past accomplishments, in general anyone who might potentially notice and appreciate your hard work will move on and be replaced by others who have no idea what you can do or have done. Careers don't just happen; they are carved out with sharp pointed implements, striking at the right time and place, itself a soft skill. It certainly helps immensely to actually possess those soft skills, but you also need to want and pursue a career to have one.
Don't go by me, though. I haven't actually HAD a career. I do smart things in the public interest once in a while, and they pay me. That's enough.