Hello Johanna,
Welcome to the forums! :qiime2:
You may have discovered these already, but here's links to the ANCOM part of the Moving Pictures tutorial and a really detailed discussion on ANCOM which covers several of the questions you ask.
Yes, you can use ancom to compare multiple different groups. Prior to that step, it might make sense to see which groups are significantly different with an anova test on a beta diversity matrix, as discussed here.
Take a look at these and let us know if you have any other questions!