If new messages don't appear in your Exchange mailbox
If certificate authentication is enabled on the mail server, make sure that your mailbox has no more than 11,000 messages. Microsoft recommends no more than 5000 messages per Exchange mailbox.
If that isn't the issue, quit Mail, then open it again and don't select any other mailbox while mail gets new messages. To be sure that Mail has finished downloading and indexing new messages, you can choose Window > Activity, then watch the progress of the download.
More Info: Use Microsoft Exchange (EWS) accounts in Mail on your Mac - Apple Support
The Inbox syncs more frequently than any other and should be kept as clean as possible. (this is the number one cause of problems with sync). While Microsoft and Apple gives higher numbers, you'll find that smaller Inboxes do not have the problems with Sync.
Do NOT make subfolders under the Inbox. It's not prevented, but it will definitely affect performance. Just like they let you rebuild an Identity with Exchange it's not recommended. If you ever have a problem with an Exchange account, you should remove the account and let it sync back from the server.
Make folders at the root level.
A subfolder <- do not do this
A subfolder <- do this at root
B subfolder <-this is OK
Any folder with more than a few thousand messages is going to take some time to fully come down when you first sync your account. Due to how exchange works, you will see the 512 newest messages in a folder at the initial start of sync then the rest of the folder's contents will be back filled before you see any mail that arrives after this point. This is why it will look like you are not getting new mail when a folder is still going through initial sync. This is also further exacerbated by an Exchange issue where it gets unnecessary change events before the newer mail arrives. If you carefully watch the progress during the sync of a folder with greater than a couple thousand messages, you will see "Updating local..." a lot before the newer mail starts arriving (newer than the initial 512 that come down).
The Inbox does get high priority so it will generally sync before other folders that also need to sync.