tl;dr a lot of the things that are coming out here make it seem like we're looking for advice on how to undermine one's line manager. One should, generally, work with their manager and rise with them. If the manager is a toxic person, and actively try to suppress the employee, then change managers first.
I take the initiative on a project and I want senior management to know about it. My manager might want to be the one making the announcement about HER/HIS TEAM achieving something great.
The manager should be the one making the announcement to senior management. Maybe they'll talk about specific contributions if they were instrumental, maybe they won't. It's a team achievement, but some people can get shout outs for individual excellence.
I want mentoring or coaching from senior management. They might say "you should ask your line manager", or they might be happy to help, but then my line manager would want to know what we have been discussing.
People can often talk with their manager looking for advice on finding a mentor within the senior management team. Generally, it's better to find people who are one or two years and a level or two above you since they're still the most in touch with the level the mentee-to-be is at.
I might want to discuss my big ideas with other managers, either more senior or in other divisions. My manager might feel sidelined.
Depending on your work culture, one can probably go ahead and talk to managers lateral to their current manager. It's definitely possible to have negative feelings from the current manager because the current manager is 100% being sidelined. Senior management is also pretty tricky because "big idea" people are everywhere. Everyone thinks they've got the next big thing. Execute on ideas, build a proof of concept, do the numbers, make sure it's attributable to oneself, and then work with one's manager to make those ideas come to fruition. If they're so good, get out and become a senior manager by starting a new company around that business.
I heard about things like "ask for permission later", but I am still confused.
In much on the US, people can be fired at any time, for any reason. Best to get people on-side before undermining them.
As this is meant to be general/universal advice, how does it happen in reality? How does one build relationships and get visibility above her/his line manager, without damaging the relationship?
Generally by getting the manager on-side and having them become a "sponsor" for the employee's work. The visibility comes when the managers starts talking about the employee because the employee is doing a great job, and puts the manager in a great light.
And if this introduces inevitable friction, how to handle it professionally? I want to move up, but I also want to be as professional as possible about it... and my manager and I might have different ideas about my progression.
It shouldn't produce any friction unless the employee goes about it in such a way that they make it a problem. One should be working with their manager on the path forwards and upwards. If they refuse, look to another department, or look to change companies with a step up.