Glad to hear you were able to retrieve your emails. The web browser method is very useful as I find that Mail holds only a month or two of emails, whereas via the browser I can go back to all (undeleted) emails.
Wrt ‘pulling down’, the search box is there, it is just not immediately obvious. Go to your Inbox in Mail. Place finger lightly on the list of emails and slowly slide your finger down. The search box should now appear directly under the title Inbox. When you press the search box it changes to give you a text entry box and the option to list unread, flagged, VIP and attachments. Additionaly there is the option to select current or all mailboxes.
Btw, a faster ‘pull down’ on the list forces mail to check your server immediately rather than wait for the next auto-check.