Why don't emails from Odoo show up in your Outlook Sent Items?
Because Odoo sends them itself, outside your mailbox. Outlook does not know about them. The mail only lives in the chatter of the record in Odoo.
As long as you work in Odoo yourself, you barely notice. It hurts the colleague who lives in Outlook: they see the customer's reply come in, but not what you sent before it.
Can you sync emails from Odoo to Outlook?
Not out of the box. You can BCC yourself, or set your mailbox as the outgoing mail server. Both only half solve it.
A BCC gives you a copy in your inbox, not in Sent Items, and you have to remember it on every mail. With your mailbox as the outgoing server, Microsoft does send the mail, but whether a copy lands in Sent Items depends on how your provider handles SMTP. Test it with one mail before you rely on it.
And the other way: how do you get mail from Outlook into Odoo?
Odoo has an Outlook add-in for that. From Outlook you log a mail onto a contact or opportunity in Odoo.
It works, but it is manual, one mail at a time. Whoever forgets leaves a gap in the customer's history.
How do emails from Odoo end up in Sent Items?
By sending them through your mailbox instead of through Odoo. That is what Mail Pro, which we build, does: Microsoft or Google sends the mail from the mailbox of whoever sends it, so afterwards it sits in Sent Items with everything else.
Inbox and Sent sync both ways every minute, and replies land on the record they belong to. Mail Pro works with Odoo 19, Community and Enterprise, on Odoo.sh, Cloudpepper or your own server. Not on Odoo Online. The beta opens in October.