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/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 for Windows and above..
EAS - Another Microsoft protocol that is the industry-standard synchronization method for mobile email clients 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. EAS is no longer supported by Microsoft for syncing to any other desktop client. (E.g., Microsoft Outlook)
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, does mailbox size matter?
Generally, no. That said, even when syncing a desktop client, some time constraints should be put in place. For example, only syncing a year of data from SmarterMail to the client, not all data. Beyond that, the webmail interface can be used and is much more stable. That said, MAPI/EWS can handle larger mailboxes with little issue. If neither of those are available, IMAP is the protocol you should use for your email. For your calendars or contacts you can use CalDAV and CardDAV, respectively. IMAP has push-like support with the IMAP IDLE process, which pushes new messages to your desktop client at a set interval. (MAPI/EWS is essentially a real-time sync, meaning messages are pushed to the client as they come in.)
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. However, some clients now offer the ability to sync a year or more. We do not recommend syncing that much data based on device limitations, connectivity issues, etc. For a more consistent experience, mobile syncs are best limited to a few months of data.
When I do use MAPI/EWS with Microsoft Outlook 2016 on Windows or Outlook for Mac, does it sync calendars and contacts as well?
Yes. It will automatically start pulling down all items in Calendars, Contacts, Tasks and Notes. 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 Mail does it use EAS?
Yes, if you choose the "Exchange" option when setting up your account, Windows 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.