There are no rules here, just experiment and see what sounds best! One thing worth mentioning however, is that if you are trying to make a singing synthesizer it is important that your voice follows the rhythm of the instrument at least fairly well. If it does not, your output will most likely be disappointing.
Another word of advice, do not vocode your voice with a carrier that contains drums if you are aiming for a clean result. Drums will make the output sound very boomy and this makes it extremely hard to hear the words of the modulator. Also, if it is important to you that people hear exactly what the vocoder is saying, be sure to speak slowly and very clearly.
No, you don't. It is the vocoders job to make sure that you are perfectly on key. If the carrier is on key and the modulator follows its beat, you'll sing exactly like the carrier plays. It doesn't matter if you are one octave above or below, or if you can't hit the notes right; you will still have perfect pitch. Note however that a vocoder is not the same as an auto-tune filter or a vocal harmonizer. A vocoder does not in any way aim to create a natural vocal sound, it is only used as a synthetic effect.
There are a number of reasons why the output signal might sound bad, or not as good as you expected. The most common cause is the use of an unsuitable carrier. You should try to use a carrier that is continuous, to prevent the output from sounding choppy. A string section usually produces a good result, and so does a brass. I even tried with a synthetic electric guitar once and got some surprisingly cool effects.
If you are confident that your carrier is not the cause of the problem, then I suggest that you try and tweak the parameters a bit to see if the quality improves. There are many factors that control the vocoding process, and a slight tweak may be just what your audio needs.
99 % of the time, the answer is no. Since a vocoder is completely monophonic and the mixes of today are almost always stereo, you'll need to tweak the sound a bit before you can insert it in your mix. Here are some general tips:
There are situations where an unedited vocoder signal might actually suffice, for example if you have a section in your song where all the instruments go away except for a few really soft ones, you might find that the raw vocoder output creates a really interesting contrast there. But even then, a slight delay would probably improve the effect even more...
This greatly depends on the style of music that you're making, obviously you don't want to use a vocoder in a slow, tender ballad. Vocoders are mainly used in Dance and Techno music, but of course you're free to experiment. Just be careful not to overdo it, there is nothing worse than an entire song with only a vocoder doing the singing. Using it too often will make you sound like an amateur who just found a new toy. Use it to create effects, not for the whole vocal track. Using a vocoder in a computer game can also be great fun, especially for robot and alien voices. Again, don't overdo it there either.
The short answer is no. Even if you don't have a 500-dollar microphone and a 2000-dollar mixer, you'll still be able to use the vocoder with good results. Even a 50-dollar headset will work, although the sound won't be as good as with a higher quality microphone for obvious reasons. Basically the better your equipment is, the better your final output will be. However there are many ways to hide the fact that you may not have the best recording setup in the world. Mixing music or sound effects on top of the vocoder output for instance, will effectively blur out the background noise in most cases.