Migrating email from current email provider to Microsoft O365

We would like to confirm our understanding of how to migrate emails from a current email provider to O365.  Users all have Outlook 2013.  These are our anticipated steps:

1.  Procure the O365 accounts.
2.  Add the O365 accounts to Outlook.
3.  Change the MX record at the domain registrar to point to the new email provider (O365).
4.  Export emails and contacts from the old accounts  to a .pst file.
6.  Import the .pst file to the new accounts in Outlook.

Question 1:  It is clear to us that Outlook will now contain the old emails, and will start to receive the new ones... but how does the O365 web mail server get those old emails?  Will they sync up automatically?  Or would we need to use the "Network Upload Method" we have seen referenced.  (In general, unless we can be convinced otherwise, we would prefer using methods that are more manual than highly automated, and perhaps more technically complex.  We only have a few mailboxes to migrate and they are small in size.)

Question 2:  We intend to move the domain registration from the current registrar to Microsoft.  Would we be correct to assume that we should do all the email boxes first, wait for the dust to settle (a day or two), then just move the domain registration to Microsoft?

Thank you.