The 1200 watts is the RF output, right?
Typically, it's the High Voltage transformer's input, but the RF conversion in microwave ovens is relatively efficient, which you can see by the fact that they don't burst into flames very often.
If I opened up the microwave and take out the magnetron, could I use it as a radio?
You need a magnetron, the excitation electronics, and then something to actually modulate the output of the magnetron with information. Additionally, you'll need output power and frequency stabilization that domestic microwave ovens don't need.
Magnetrons are, if you will, just a RF cavity resonator that can be "pumped" with external high voltage. They need to be operated in a continuous mode, at relatively constant power.
That means that your method of getting some information on the generated RF needs to deal with the high-power signal as is.
Therein lies the technical challenge!
When you just switch your magnetron on and off, you'd get terrible spurs due to the process of getting your non-radar magnetron up to power. You'd need to account for that by more high-power RF switching and filtering, which is where you quickly cross into "more expensive than a good microwave amplifier".
Generally, you can't switch a relay with a microphone. A relay can be switched a few times per second, and will wear out after a couple million switchings, whereas audio has more than "on/off" information, and also crosses zero a couple thousand times per second. So, nope, this is architecturally not possible at all.
According to my research, the IC-905's output power is only 10 watts, so would 1200 watts be too much?
If you want to fry birds in-flight, it would be appropriate. I don't know any communication signal standard that would use microwave frequencies and require that much power from an amplifier.
I didn't get any useful results on if hams can operate on 2.45GHz, but I'm guessing not.
You've got the KC3WCR call sign, and that means you're in the US; and as you know, legal bands are subject to local legislation! So yes, you can (that is extremely simple to research!).
Can I change the frequency of the magnetron so it operates on, say 10GHz?
No. The range of resonant frequencies is defined by the shape of the magnetron.
Also, 10 W is very significant output power; I'm not quite sure what your motivation is going for 1200 W? You would literally damage a lot of devices in your vicinity, you would probably cause bodily harm to yourself, and you seem to be vastly underestimating the cost and complexity of even transporting 2.4 GHz, let alone 10 GHz, at these high powers for any significant distance.
Note that it makes little sense to say "I want high power, but I only want to transmit voice". What would you need high power for? Free Space Path Loss at microwave frequencies means you're not doing far links, anyways: Adding a factor of 10× in power is very expensive, but only means a minor range increase. The high bandwidths you can use in the microwave bands mean that you can take your voice information, and spread it in bandwidth as needed, so that you can work at low spectral density. "More power == more good" is a falsehood that's sadly very common in the ham hobby, but in reality, systems need to be designed for transmitters to use as little output power as possible – otherwise, you just end up in a screaming match where everyone in range tries to scream louder than all the others, leading to overall worse performance than if everyone was using only little power.
