Received emails disappear immediately

Question

Once a user complains some emails are directly delivered to the Deleted Items folder instead of Inbox, what should we do to troubleshoot the issue? Some of the common reasons are listed below.
Rules
Both Outlook-client and server-based rules may have deleted some specific emails, once these emails meet certain conditions, they will be deleted automatically.
On Outlook, please check whether there are any rules with the action “delete it” or “permanently delete it”. We should note these rules with the action “move it to the specified folder” may also delete emails if the specified folder is Deleted Items folder.
Remove the rule or change the conditions/actions to stop deleting these emails.
On the webmail, we should perform the similar steps to make sure there is no such rule that deletes emails automatically.
Junk E-mail Options
It is possible that the Junk E-mail setting has deleted the email permanently, please check the Junk E-mail Options:
Once “Permanently delete suspected junk e-mail instead of moving it to the Junk E-mail folder” is selected, emails that are considered to be junk will be deleted permanently and won’t be delivered to Junk E-mail folder, especially when the level of junk
e-mail protection has been set as “High”. Anyway, we can’t guarantee normal emails won’t be considered as Junk when it has been set as “Low”, the filter can’t be 100% accurate.
Default delivery location has been set to a .pst file
For some Exchange Accounts, we may not want to keep the emails in the mailbox, instead we need to keep them locally. We can set the default delivery location as a .pst file, which won’t leave a copy on the server then.
To check the setting in Outlook 2010:
File -> Info -> Account Settings -> Account Settings -> Data Files tab -> Select the Exchange Account and make sure it has been set as default.
Then on the E-mail tab, select the Exchange Account and verify the delivery location is set to the correct mailbox.
Note: The feature to change a default delivery location has been removed Since Outlook 2013.
Ignore Conversation
The feature “Ignore Conversation” can stop the emails of a conversation to disturb you if you are not interested any longer, the entire conversation will be in the Deleted Items folder. You need to go to the Deleted Items folder and select any message of
the conversation, right click on it and select “Ignore”.
Move Conversation
The feature “Move Conversation” will always move messages belonging to a conversation to a specific folder. If the specific folder is set to Deleted Items folder, it makes us feel like it’s been deleted.
To stop moving a conversation, select a message from that conversation, right click on it, select Move -> Disable Always Move Conversation. This will not move back the entire conversation to the Inbox folder, however, new messages will be back to the
Inbox again.
Mobile Devices
To work efficiently we always have our accounts setup on multiple devices nowadays, such as Smartphones and tablets. If they have downloaded the emails via POP protocol, emails may get lost from the server. If you have your email account setup on multiple
devices, please consider to check on them.
Anti-virus integration
If you are using an anti-virus program and it has some integration with Outlook to scan the incoming emails, it may work like the Junk E-mail Settings that delete emails. Please check the related settings and disable its add-ins to verify if it is the cause.
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Answer 1

http://kb.mozillazine.org/Disappearing_mail