I've got a 5s however upon researching online it seems the iPhone will only sync the Inbox automatically and not any self created folders. I've got a few simple filters setup so emails from Amazon will go into a eShopping folder etc.

Are there any email apps that I can download from the play store which provide the functionality to sync all the folders?

Kind Regards

you do know if you click mail on your iphone.. then before you go to your email account scroll down and your account is also listed again with access to your folders

Have a look in settings-> Mail -> <your mail account> -> Push and see if you can turn on the folders you want synced.

iPhone Mail otherwise only syncs a folder when you tap on the folder in the mail app.

I use push as I use an Exchange account (corporate) rather than IMAP.