There is a good chance that the Sent emails will just show up eventually, after taking a short vacation on the server before returning home. Most Sent folders are on the server, meaning sent emails have to make a round trip before they show up on your iPad (or other devices).
If you go to Settings > Mail > Accounts > (Choose Account) > Advanced you can see/set what folder are used to store Send, Trash, etc. Make note if the folders used are in your account or local folders. Local (On my iPad) folders will not be synced across devices.
For iCloud Mail you need to drill a bit deeper to Settings > Mail > Accounts > iCloud > iCloud (and scroll to bottom) > Mail > Advanced.
Google does some non-standard IMAP things, which sometimes gives non-google email apps fits; Mail include. Though mostly it works. I don't see a way to assign a Sent folder.