if these questions are set for everyone to discuss, I'd say:
Admin can run there servers how they like?
Yes in theory, but practically no. They pay for their servers so they rule, but they are running a game which has it's own TOS, so you can run your server how ever you like but you need to follow certain rules, also using common sense is allowed
Are they allowed to abuse verbally?
No one should solve their personal grudge between others with admin powers, then again abusing admin powers shouldn't be allowed. This is internet so you can't completely cut trash talk off, but sometimes shit hits fan way too hard and you get huge mess. Some takes the lightest trash talk very personally and some doesn't care if it goes a bit dirtier, everyone has their own level what needs to be reached before they get fed up. We've seen everyone that some of the worst trash talkers can't see themselves when they go too far but it's funny how they are so butthurt after they get some trash back. Also it's markable that majority of the community doesn't care until the shit hits on themselves, they just watch aside while someone is being bullied hard.
Basically if you can't stand the heat - stay out of the kitchen, but then again it's not that simple. Every one of us are different kind of persons. I'd like to call again for common sense, but I think I'm asking too much...
Are they allowed to TK?
No, teamkilling is forbidden and the same rule stands for all. Sometimes it happens by accident, sometimes it's intentional. That's why playeradmins exists so they can sort out who's teamkilling on purpose and those players can be banned. Playeradmins should be responsible and sensible players who knows what's allowed, what's not and what they need to do. If playeradmins starts abusing their rights and starts teamkilling at their server you should talk with server owner and show printscreens as evidence, maybe they rethink their roles in administrating the server. I haven't played many fps games but those I have are having their own rules for teamkilling and it always results you something.
Basically these what you're asking are handled by server administrator. They set rules how you're allowed to act at their server and if there's someone teamkilling, they make sure it results a ban at their server. When problem goes outside from private servers needs Assist Admins to take action, but as far it happens inside private servers it's really server owners job to handle it. Don't take it too literally, if there's massive teamkillers causing havoc at couple of servers then bring it up at forum so Admins can also have a look at it.
Very nice post, Ronski!
I'll bet that Vegeta will disagree with at least something. If not with what you said, then with something I will say.
Like Ronski said, you can run your server as you wish up to an extent.
Your liberties end where other's begin doesn't hold here literally (you have no obligation to share a server you paid for), but I think we should still try to uphold the principles.
If you want a server with very specific set of rules, you can protect it with a password and have only those join that you want to be there.
The second question is pretty sill, to be honest. It's sort of like asking "am I allowed to drive like a maniac?" - you're not allowed to do it, but if you're not hurting anybody, you might get away with it for quite a long time. Why do you feel you need to be able to verbally abuse people? To feel superior?
I know trash talk does happen, and it's fine to an extent, but I just don't know what kind of answer you're looking for here. Take away the word "verbally" and try answering the questions yourself, if you don't see that's not right then I can't really help you.
No, teamkilling is not OK, from the top of your post, I thought you knew that? As I mentioned earlier, if you want to create a server where "TK is incouraged", then protect it with a password, invite your friends and be happy.