There is some confusion over the various types of synchronization options that are available for desktop and mobile email clients and devices. To clear some of this up let's first look at each protocol.
MAPI / Exchange Web Services (EWS) - Both are protocols introduced by Microsoft for Exchange that are intended for desktop email clients such as Microsoft Outlook. At this time, Microsoft Outlook 2015 for Mac from Office 365, Outlook 2011 for Mac, Apple Mail and eM Client support EWS. MAPI is supported by Outlook 2016 -- O365 versions as well as standard desktop versions -- for Windows.
Exchange ActiveSync (EAS) - Another Microsoft protocol that is the industry-standard synchronization method for mobile devices such as iPhones, Samsung Galaxy and Note phones as well as iOS and Android tablets. EAS is also available for use with Windows Mail, People and Calendar and the Outlook client for iPad.
IMAP - One of the most pervasive protocols for email retrieval, alongside POP3. IMAP is cross-platform and can be used by desktops and mobile devices. IMAP is extremely stable and reliable and is a perfectly acceptable way to synchronize your email across all devices.
So, what protocol should I use based on my mailbox size?
For your desktop, if you have a large mailbox, OVER 5,000 messages or 5GB of space used, IMAP is the protocol you should use for your email. For your calendars or contacts you can use any other of the sync protocols (such as CalDAV and CardDAV) but for your email, you should use IMAP. IMAP has push-like support with the IMAP IDLE process, which pushes new messages to your desktop client at a set interval.
If you are on a mobile device, EAS can be used, even with large mailboxes. The reason it works well on mobile devices is because mobile clients only ask for a set timeframe for synchronization, like a month of mail or less. It will not pull back all mail in your mailbox unlike Microsoft Outlook 2013 for Windows.
When I do use MAPI / EWS with Microsoft Outlook 2013 on Windows or Mac, does it sync calendars and contacts as well?
Yes. It will automatically start pulling down all items in Calendars, Tasks and Contacts. However, please see the guidelines above for which protocol to use when synching your desktop client to your mailbox.
When I set up an account in Windows 8 Mail does it use EAS?
Yes, if you choose the "Outlook.com" option when setting up your account, Windows 8 Mail uses EAS by default and so it also syncs People and Calendar. Again, please see the guidelines above for which protocol to use when synching your desktop client to your mailbox.
If I don't have the EAS add-on for SmarterMail what do I do on my iPhone?
You may want to use IMAP for your email and CalDav and CardDAV to bring down calendars and contacts, respectively. This would all work with an Enterprise Edition of SmarterMail.