You could use lavs for some types of live performances (especially if you just want to record, but not necessarily amplify), but you're going to run into some major problems the minute you're working with high volume.
Lavs are generally omnidirectional mics, which means they're going to pick up sound roughly the same way from all sides. This is the natural enemy of a feedback-free performance, since you'll have a much harder time controlling for the reflected sound from your PAs. For this reason, when you see wireless headset mics worn by public speakers, singers, etc., they're almost always directional vocal mics, not omnis.
There's a reason that you usually just see a very directional mic (e.g., cardioid family) used on-stage for loud music shows. Just makes your life easier by cutting out sound from directions other than your intended source.
Yes, so long as there's basic compatibility between mic and transmitter, i.e., input jack type / threading if applicable, voltage, etc.
Different mics are...well...different. Whether it's so noticeable that you have to make changes is a question for you to figure out. One thing that can help if you can't tell with your ears is to take a look at either a spectrogram (which shows you frequency distribution in a visual way) or look at the frequency charts for the different mics and see if they're markedly different. Let me know if you want more info on this!
Keep in mind this can actually be a problem. Again there's a good reason for face mics in a performance setting. If you turn your head significantly, there is no mic placement with a lav that's going to give you good sound throughout the entire range of head movement. A face mic stays with the mouth at all times and gives consistent tone and volume.