Can I get my Galaxy 4 to put sent emails on my server?
How have you configured your emails? Is it POP3 or IMAP? Yahoo have a guide to explain the differences. I'd recommend using IMAP as it should sync read status between phone/webmail. You can find the info and steps needed here.
As for sent emails there is probably a setting to sync sent items (or you need to put them into the correct folder). If you provide more details about the app/how you configured it we can help further