Hello ..... I have Outlook 2003 on my DeskTop which I work from when in my home office --- I have Outlook 2010 on my laptop which of course is my work station when I travel. I also have my account on my Android.
My e mail address is from the earthlink.net domain.
If I understand correctly you are unable to download some emails onto your laptop or cell because your desktop has already downloaded them and removed them from the server.
I assume you are using a pop3 email account so instead of setting your desktop to remove data from the server after downloading emails you could set it to leave a copy of messages on the server for a set time say 14 days. This will give you time to download emails onto all your devices before they are removed from the server.
To do this In Outlook 2003
in Outlook 2010
Click the File tab > Account Settings
Click on account settings again then follow from step 3 above