- A good manager will notice, and appreciate, your effort. But promote you? You are one of his best tools, why should he? Usually he'll try everything he can to keep you where you are. If it's a bad manager, he won't even recognize your effort.
- Yes, unless you work with mixed teams or on different projects. In that case, two managers might end up fighting to get you on their teams and it might be a good thing for you.
- Nothing to say here, you are totally right, but depending on how you play your cards, the reward can come earlier. Only thing I'll point is that I've seen departments go to hell because managers failed to recognize this.
My tip would be to communicate. If you have free time, don't wait for your boss to assign you work and ask. If you see something else that you can be doing (sometimes there are features or bugs that could be worked together), say so, but always comunicate first and wait for your managers decision.
Of course, all this is IMHO and your results may vary.