Oh don't quash their trekkie nerdrage with such things as fact about copyright, trademarks and the obligation to stop infrinctions to keep your rights.
Oh, CBS/Paramount had every right to step in. What's unfortunate about this whole debacle is that they could have protected their copyright and still earned the good graces of the fans. Because, well, they may own the IP, but it is the fans that make it profitable. Drive away fans, and it will won't be profitable to put any work into the franchise.
I don't really care about the Axanar movie one way or the other (nor do I really care about any of the other fan productions). But, it should be noted that up until this point CBS has given tacit approval for all of them. And in the case of Axanar, the team behind it had been in contact with multiple people at CBS (including executives) and all such contacts had been positive. The Axanar crew had even offered to pay any "profits" made to CBS. That was refused. Even JJ Abrams was trying to get CBS to drop the suit.
Anyway, the heart of the matter to me isn't whether Axanar went to far (which they clearly did, and have acknowledged such), but rather the way CBS reacted with their a"guidelines"; so how does one protect your intellectual property without alienating the fans that buy it from you. Because yes, legally you own it, but everyone that enjoys it, owns their experience of it.
Let's take another example: Topps sends a C&D letter to Amazon to take down all of the unlicensed BattleTech fiction from the Kindle store (yes, there actually is quite a bit on there, don't believe me, I'll PM you links), which is quite within their rights and duty to do so. But on top of that, they then say that if you want to write BattleTech fanfiction it can be no longer than a single paragraph and you are not allowed to upload it anywhere online. How quickly will that alienate the fans?