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I use K9 setup with 3 different IMAP accounts. I don't know if this satisfies all of your requirements (don't know if it allows you to set up automatic filters or not), but it does support IMAP folders, and with a little configuration, gives you multi-select checkboxes in order to move/delete email in bulk.
Well, I've kind of got used to the forum text editor on my phone, so I'm back.
Just to say that MailDroid also supports automatically moving incoming messages into folders based on the subject and sender.
So you can have your emails handled and have a rich text editor for composing.
But there is no calendar; it's purely an email application. I suspect you will have to use two or more applications to achieve your goal, but MailDroid should do what you need as far as email is concerned.