Hi @arlandan,
There are a few other discussion here here, here, and here, that might give you some in-depth insights into the questions you have here. There are likely more on the foru, but those ones I found the fastest.
But brief answers to your questions:
This really depends on what it is you are doing. For some actions, like alpha group significance or beta group significance, you should, but for tools that are compositionally aware like ANCOM, ALDEx2, songbird, you shouldn't.
Normally I would say yes, as long within your experiment you can justify this being an outlier. For example if these were 11 mice from the same treatment, then there is reason to think something happened to this 1 mouse and it is clearly an outlier and so you could justify removing it. In your case however, I am a little concerned with regards to your sampling depth. A depth of 700 sequences is not enough (for most common microbial communities we see anyways) to really capture the full diversity. I Would recommend re-running this with a much higher max depth, say 10-20k at least. I personally go even higher but the benefits diminish rapidly thereafter.