Thriving in a competitive lab
From what you are telling, your advisor has not bred a competitive lab but rather a horde of children. The behavior you are describing is unacceptable and shows no respect for each other at all. It normally is the professors job to make clear how the lab is working, to make sure the speaker in a talk is listened to and discussions are postponed to a break or after the talk, etc.
Furthermore, if everyone in the group is competing over a task, this shows that they don't have enough tasks of their own and go for everything that is available. Once again, it is the professors job to have regular discussions (in private!) with lab members about their current projects, to make sure that they go along well and that such a case as you described in the last point, i.e. everyone has nothing to do, does not happen.
All in all, it looks like this lab you are describing has no real leader, no one capable or willing to set ground rules and advice and guide everyone. I think and hope this is not the common case in labs and would advice you to avoid such a place if possible; maybe you are able to find a new advisor?
Side note: Of course competition will always be a thing and a healthy competition is good. But what you described is not healthy at all and in all the labs and research groups I visited so far (math, CS and engineering mostly), there was a decent amount of respect for each other.
In competitive lab members challenge each other.
In healthy competitive lab each member has their time to present and defend their work. Competitors show respect to each other.
In unheathy competitve lab each member intrigue againts their colleagues to grab attention.
In your case, even worse, members do not show enough respect to each other to listen their presentation!
Run. Actually, you should have your things packed yesterday and be knocking on different lab's doors.
Your adviser has not created a competitive culture, but as pointed out by other responses, a toxic one. Unfortunately, this case you describe is not rare at all, so don't despair. Basically, the circumstances you are facing are not an exception. They do occur.
Why does this happen? Because your adviser has taken the easiest route: inaction and lack of confrontation. Inaction leads to self-management of the group and situations like the one you described, where bullies thrive and reasonable hard working "nice" people get resentful.
Advice? I would suggest you find a new group, because this is an unsolvable problem: it's the adviser who creates and maintains a culture, either by action or inaction. Furthermore, changing to another research group is usually hard and you have no guarantee that the next one will be better.
As difficult as it is, think that the person that should be happy with yourself is you, not your supervisor. Do you want to look like a smart-ass or become a bully to get his attention and approval?
Also, learn to defend your territory. Don't let anyone bully you, and tell them off if you need to. If you need to say "do you mind withholding the questions until the end of my talk?", just say so.
Regarding the "success or influence" you talk about, how do you define "success" exactly? Success is defined by your expertise, skills, knowledge and publications, which are independent of what your adviser thinks of you.
My condolences, and good luck!