I resolved my issue.
I just let you known the fix in case of others falls in the trap.
1) I had to know that the mailbox of the Internet Provider was hosted invisibly by Gmail
2) I had to know that Gmail allows only ONE download for each email. After the uniq download the email is flagged as downloaded and nobody else could get it again except from getting connect to WebMail Gmail
3) I configured my Windows Live Hotmail to retrieve the emails from this email adress
4) I configured as well Windows Live Mail to retrieve the emails from this email adress
As Windows Live Hotmail retrieve email every 30mn (PS: max 50 emails each time), when WLM was not running (PC in sleeping mode) all the emails were downloaded to Hotmail. When WLM was running and was retrieving email every 10mn everything went pretty good except if the hotmail extraction occurred just between 2 WLM extractions thus stolen eventual emails from WLM.
I removed the reception of the mails from my Live ID of Windows Live Hotmail