Am a novice when it comes to all things outlook. I had a similar problem, I would receive an email on my smart phone but once I accessed outlook and opened the same email/marked it as read it would automatically disappear from my smartphone(once you read from outlook the email server marks the email as read making it inaccessible to other devices).
After a lot of playing around with settings, I stumbled upon a solution that worked for me maybe it may be of help to you.
1. Open outlook(am using 2010), go to file then info.
2. Under info, click on account settings(click twice)
3. Double click your email account name.
4. Click on the "more settings" button, its on bottom right corner.
5. Go to advanced tab. Near the bottom of the window under delivery, check the leave a copy of messages on server option (it is unchecked by default).

Hope I was clear. Good luck.