So... taking on the specific questions in an internal company context... there's a fair bit of "it depends on the situation" here because you may have wildly different relationships with groups, even within your own company. I definitely change my tone and approach depending on my audience - both the nuances of how my team will be working with the other team, and also the rapport I have with the other person I'm speaking to and the degree of privacy involved.
If someone achieved something great in the team, it is thanks to me that he was actually enabled to that process; I hired him, trained him, and gave him the instruction, etc. Do I lose the limelight when I become a manager?
You loose the limelight in the same way that the coach of the team looses the limelight in a sporting event. During the game, the announcers are and should be talking about the awesome (or abysmal) actions of the individual players - the slam dunk, the great pass, or the awesome tackle. It's not "wow, what a great coach, he really taught that guy to tackle", it's "that player did a great thing". For the coach to run onto the field claiming credit makes the coach look a bit cheap.
But - in between games, when the moments of glory are over, the coaches get talked about. Listen to any long sports discussion segment and it's about what the coach is doing with the team. Did he make a great hire, who knew that this player would be so good in that position? What great coaching! Generally the coach doesn't have to sit there and toot his own horn, the play of the overall team is the judgement of his worth.
It's just a different limelight. And not one that is quite so directly shown upon you. A lot of the really good feedback about a boss comes through behind your back. When people love or hate their boss, it comes through to the higher level management as praise and complaints that you won't necessarily hear.
How am I supposed to showcase my performances and achievements to the company, in order to gain visibility and get promoted to the next step - if everyone else is doing the actual hands-on work?
The same way you did before. Look for challenges that you and your team can take on. Now you have a bigger scope to accomplish them, because it's not your work alone. See a problem? Suggest that you tackle it with your team. Do point out the productivity of your team, and talk about things that you (as a team) are doing to change it.
If any one individual takes credit for something that went wrong,
wouldn't that put them in danger of being blacklisted from the
potential promotion candidates?
I don't agree with taking undue blame, I do endorse taking responsibility. You can't know everything or fix everything. If someone has acted against your direction - say so and let the person take the blame. If the mistake is truly a mistake from a bad process or poor team communication, say so, and propose a better way. That's taking responsibility - not blame. From time to time, you may also be in the position of saying "fixing this sort of problem is not worth our time, and I will take responsibility for that decision on a large scale and any bad impacts... because I'm also taking responsibility for getting the right things done on time and on budget."
The individual mistake (one guy making an error) is not your fault, but if your team makes the same error over and over again, having a team that can't change its ways is your problem the same way that having a team that makes success after success is to your credit.
What does the rest of the company actually care about the specific
individuals - whether it was something good or bad?
Most teams don't work in a bubble. Individuals from your team work with individuals on other teams. Those individuals care very deeply about finding the right guy for the right action.
Also - teams change - your team won't be the same people in two years - I can almost guarantee that you will loose or gain people as the team morphs with the company. Other managers and individuals in the company are actively wondering if your team members may be good for their teams, or if they can get into your cool team when an opening is available - who your team are as individuals and how you treat the individuals on your team is very much a critical factor here.
More importantly - the company is the entity paying these people - not you. So someone somewhere is trying to figure out how much budget your staff consumes and if each and every one of them is worth it. You may have a say in how salaries and bonuses roll forth - but I doubt if you have unlimited control - so in addition, at least a few others in your company care about knowing the value of the individuals on your team, and assessing that in monetary terms, both within the team and across teams.