I am biased, but I am on just to answer your question on push. Push has two definitions:
1. True Push: This is blackberry style
2. Idle Push: This is available for most IMAP servers, but works differently than true push.
K9 and MailDroid both use Idle push (I never heard of that other mail client). True push you won't really find in mail clients on Android (I believe Seven does some form of this, but can't vouch for it or how good it is).