The disappearance of messages from the Inbox after a month suggests that there is a device connecting to the mailbox using email software that is set to delete messages from the online server after one month / 30 days. This is not unusual for POP3 email account software.
So the question for you is, do you have any devices connecting to your mailbox that have been set up as POP3 accounts. If not you then it would be wise to consider that your email password is compromised and a rogue device is connecting.
Scan your email devices to remove viruses, trojans and other malware and then upgrade the password. I recommend 12 - 15 characters, multicase letters and at least one number and one approved symbol. Check also in the online mailbox Mail settings that there are no Rules operating to spy on your messages and in the Security section that there are no unrecognised devices connected to the mailbox. Sign out all if in doubt.
