Man lernt nie aus - as the German say - one never stops learning. Thanks to comment found on my blog I found out that there is very nice solution problem of disappearing emails (after you download it with one POP client, other POP client would not get it). It is even documented in GMail help:
http://mail.google.com/support/bin/answer.py?answer=47948&topic=1556If you're accessing your Gmail using POP from multiple clients, Gmail's recent mode makes sure that all messages are made available to each client, rather than only to the first client to access new mail.
Recent mode fetches the last 30 days of mail, regardless of whether it's been sent to another POP client already.
If you sign in to Gmail using your Blackberry, you're signed in to recent mode automatically. For all other POP clients, replace 'email@example.com' in your POP client settings with 'recent:firstname.lastname@example.org'.
Thanks to unknown reader who made me aware of it.