I have recently been having the same problems, Although mine were an abrupt change after having changed from previously using BlackBerrys, Droid-X and most recently HTC to using the new Samsung S4. I have recently (several days ago) upgraded to the Samsung 4S and use several emails Gmail, Hotmail, Yahoo and an original email that is out of the NYC area through Transbeam.com, which happens to be a POP based email server.
Here is the situation, because I do not have a lot of storage available on my Transbeam.com email server, I have my PC software (Outlook) set to remove all emails from the server after downloaded onto my computer. I use my S4 to download these emails first and then use computer in the evening to clear the online cue and save the emails I want to keep on my computer. The problem is, once this is done and my S4 then Syncs up with the email server, it removes all the emails that had previously been downloaded on the S4.
I have just reviewed this with Samsung online that there are no setting on the S4 to change this, as this is part of the current android version 4.2.2 and it does not support a time delay email erase from the device. In other words, the other phones mentioned above have a "keep email on phone for X days" which is what we need for our POP email accounts. Google and Hotmail have this sort of feature on the S4 in the settings, but it is not as necessary as the POP mail account... because the POP account does not have the amount of free storage as the other mail accounts mentioned.
The result: This will force me to return the S4

(which I like very much with the above exception) to change to the HTC ONE! All in All, it means configuring and transferring data / programs over to a new phone again and also taking several days to get familiarized with another phone and to a efficient operating space.
If anyone finds that there is an app that can correct this or if there is another email client app that can be loaded onto the S4 that retains the emails for "X" days, I would deeply appreciate knowing ASAP.
Thank you,
G