Hi @uth,
Those are excellent questions but there is no definite answer to any of those inquiries. It all depends on how much flexibility and customization you want. Qiime 2 does have a very impressive number of plugins that can help analyse a variety of simple and complex question about your experiment, but it is not as customizeable, especially with regards to graphics.
So, for example, while there is a plugin to identify a core microbiome, there's no easy way to visualize this unless you export this into another graphic environment (like R).
Characterize is a very vague term to me so I'm not sure what it is you are trying to do but bar plots is certainly one easy way to look at the composition of each sample. Check out the heatmap plugin as well. You can also create some nice ordination plots with beta diverstity and visualize with the emperor tool. Have a look through this overview tutorial diagram for a variety of options.
Yup, the core-features is the only plugin that deals with this question.
If you are looking at doing statistical testing on beta diversity matrices, you can use adonis or beta group-significance, or if you are looking at doing differential abundance testing (univariate test of each taxa/ratios across groups) you can use q2-ancom, q2-aldex2, q2-songbird, q2-corncob.
There's also loads of other third party q2-plugins you can check out over at the QIIME 2 library.
Have fun!