Interesting thread. First off, let me mention that the topic itself is all opinion. As we all know, there's no wrong way to play D&D, or enjoy D&D, or want to play D&D. We all have our various preferences which make the game fun for us. So, that said, I won't quote anyone else's responses, because I'm not trying to debate preferences, but merely add my own to the discussion.
Regarding the "antagonistic playstyle" mentioned by Reynard, I think that a true competition is impossible. Ultimately, the DM holds all the cards, so how can a true competition ever actually occur? If the PCs die, the DM has won. If the PCs don't die, it's because the DM has allowed the players to win. In either case, no real struggle can exist so long as the DM is also the final arbiter of the game.
That said, I am one of two DMs in our group, (we alternate between our two games each week). We often joke around about killing off PCs, curse when a PC narrowly avoids some horrifying fate, etc. But while that kind of joking amuses us, each of us believes that the DM's true challenge is not to kill the PCs, but rather to provide enough difficulty to push the PCs to their limits, without actually TPKing them each week.
My proudest campaign came a few years back, when I ran a game overseas. Each week, one of my players would say as she left my apartment, "Thank you for not killing us." Every week they fully believed they'd be killed. And each week, they somehow managed to come out of it alive. I never did kill off one of the PCs, and yet the feeling of lethality was heavy the entire time.
The players should feel that their PCs are not secure. They should also feel that their PCs are not pointless. It is the DM's job and joy to see that the game treads this fine line.
Regarding multiclassing, I've always treated classes as purely mechanical reflections of in-game background. As such, I have no problem with it so long as it fulfills that function. I've had wizard/rogues. I've had fighter/ranger/barbarians. Bard/fighters. And more. The in-game background is always simple "a thief who wields magic on his capers," or "a travelling troubador who picked up fencing while serving as skald on a pirate ship." Then the mechanical representation of that.
I've disallowed one-level dips for a specific ability that weren't justified by the character in-game. But on the whole, I don't see a problem with multiclassing in general. I could easily see Aragorn from LotR represented as a fighter/barbarian/rogue. It's not what's written on a character sheet that breaks the mood, it's how its presented in-game.
Regarding a DM's right to restrict classes, etc. I've always felt that a DM does have that right, but he also has the responsibility to work with the players to get them what they want to play, as much as possible. Just saying, "no, there are no ninjas in my campaign" is not enough. The DM needs to make a good faith effort to find a place for ninjas in his campaign and, if it simply isn't possible, work with the player to try and get him something as close to what he wants as he can.