Greyed Out Emails in Mail.app – How to Fix

"Sent: do NOT store sent messages on server"

Toggle this, and sent items will be stored on the Gmail server, thus enabling you access from both desktop and iPhone.

I would suggest you set all the "Store on server" options to Checked. Google's not always correct, or up to date; although they do try their best. I've never had issues with all the settings to store data on the server in the last 4-5 years Gmail IMAP has been available.

Also, look at what you have set as your Sent folder. In the list of mailboxes, scroll down until you see something like "Gmail" or whatever you named that account. Expand the list. If you see another folder in there with a name similar to "sent", it may be that your missing emails are there. Drag and Drop your known emails to this folder, then select it again, and from the menu, choose Mailbox > Use this mailbox for > Sent. Similarly for trash, junk, and drafts. I find it also helpful when dealing with an Exchange account over IMAP since Outlook on Windows XP has a different naming convention than Apple's mail client software.