I'm not clear what you mean by Stock mail - but my main email account is ntlworld.com (default - via virgin media), and yahoo.co.uk on occasions. I do have a gmail account - again used on occasions. I can access all my emails via the gmail account through the web though but is a bit more faffing about. Hope that makes sense.
You might want to check the settings of the accounts. Check leave copy on server is checked. If you use another way to check yr emails (outlook??) and that pulls in the emails then when the email app syncs they will dissapear. Dont know if this is your problem but it might be worth a check.