Just got a Samsung galaxy S4 mini (had an ancient Nokia before). When I go back to my web email (its Yahoomail), all my emails are marked as read even if I havent looked at them on the galaxy, (I can live with that though its annoying as I'm likely to miss stuff)and emails I've sent are not copied back onto the server so the only place I will ever have them is on my phone. Thats not what I'm used to - the Nokia put them onto the server so I could see them in the web application. Any idea if I can get it to do that too?
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