Some quick answers
• If by "tree structure" you mean a folder hierarchy, yep that's supported. I think we have a current bug with dragging nested subfolders but we're working on a fix. (Surprisingly >99% of users have a flat hierarchy.)
• A thread can certainly be in two separate folders (e.g. Inbox and Sent) but an individual message can't be in two folders at once. In that situation there are two copies on the actual mail server. For Gmail/Gsuite this is possible via labels where any thread can have an arbitrary number of labels. We support both systems.
• "labels can be configured to learn which emails to automatically sort into themselves, based on the email contents" -- this is a really cool idea and something we've talked about internally. AFAIK there is no cross-platform mail client that does this today beyond things like manual Gmail filters. It could also be an interesting plugin that anyone could build on NM. We have a Slack chat room where folks discuss stuff like this if you're interested: http://slack-invite.nylas.com/
• And for your bonus round (haha) yes there are 2 different ways to configure the UI. One of them is 3-pane with a reading mode like Outlook, and the other is 2-pane that navigates like Gmail. http://i.imgur.com/Lt0x7O4.png
Also in 3-pane if you make the message list wide enough it will switch into the compact version: http://i.imgur.com/SaGp9eV.png
(Obviously it will show your real mail data. We have a "screenshot mode" for sharing stuff like this without revealing sensitive information.)