Ok, from the first log, it's a connection error galore, one after another:
2013.05.13 23:03:27.558 +0200 AquaMail [NETWRK.4289] Connecting to [pop.dersie.eu:110, tlsRelaxed]
2013.05.13 23:03:32.591 +0200 AquaMail [NETWRK.4289] ***** ERROR: Unable to connect to [pop.dersie.eu:110, tlsRelaxed]
java.net.UnknownHostException: Host is unresolved: pop.dersie.eu
2013.05.13 23:03:33.357 +0200 AquaMail [NETWRK.4289] Connecting to [pop.wanadoo.fr:110, none]
2013.05.13 23:03:33.383 +0200 AquaMail [NETWRK.4289] ***** ERROR: Unable to connect to [pop.wanadoo.fr:110, none]
java.net.UnknownHostException: Host is unresolved: pop.wanadoo.fr
2013.05.13 23:03:34.246 +0200 AquaMail [NETWRK.4289] Connecting to [imap.gmail.com:993, sslStrict]
2013.05.13 23:03:34.284 +0200 AquaMail [NETWRK.4289] ***** ERROR: Unable to connect to [imap.gmail.com:993, sslStrict]
java.net.UnknownHostException: Host is unresolved: imap.gmail.com
2013.05.13 23:03:34.478 +0200 AquaMail [NETWRK.4289] Connecting to [imap.gmail.com:993, sslStrict]
2013.05.13 23:03:34.481 +0200 AquaMail [NETWRK.4289] ***** ERROR: Unable to connect to [imap.gmail.com:993, sslStrict]
java.net.UnknownHostException: Host is unresolved: imap.gmail.com
I believe this sync was "kicked off" by the device's connecting to WiFi (event based sync).
I think your WiFi connection is not quite ready (why Android sends a "connected to WiFi" event before it's fully connected is beyond me...)
Please trying increasing the "settle in" delay to give it more time.
Or, just try keeping WiFi on + use scheduled mail checking, not event based checking, as a test (besides, modern Android versions put WiFi hardware into deep sleep mode, so it's no longer a battery issue like it was prior to 2.3).
Then at 23:08, you checked mail "manually" and everything worked, the network was ready and everything's connected nicely.
Then at 23:11, the time "ticks" and my code sets an alarm for 23:21.
And then, as soon as the app starts syncing mail at 23:11, the firmware says that it's completely lost connectivity. You can see it in the log yourself:
2013.05.13 23:11:08.068 +0200 LostConnectionReceiver {}{}{}{}{} onReceive ConnectivityManager.CONNECTIVITY_ACTION: ni = NetworkInfo: type: WIFI[], state: DISCONNECTED/DISCONNECTED, reason: (unspecified), extra: <unknown ssid>, roaming: false, failover: false, isAvailable: true, isConn = false, isConnOrConn = false
2013.05.13 23:11:08.074 +0200 LostConnectionReceiver {}{}{}{}{} onConnectionLost, EXTRA_NO_CONNECTIVITY is set
And then we go again with a full round of connection errors:
2013.05.13 23:11:09.993 +0200 AquaMail [NETWRK.4467] Connecting to [pop.dersie.eu:110, tlsRelaxed]
2013.05.13 23:11:10.062 +0200 AquaMail [NETWRK.4467] ***** ERROR: Unable to connect to [pop.dersie.eu:110, tlsRelaxed]
java.net.UnknownHostException: Host is unresolved: pop.dersie.eu
So, to reiterate, your device's connectivity to WiFi is unstable, or the firmware is lying that it is (the "lost connection" event).
One other suggestion I have, since you're running 4.2.2, is to go into the device's WiFi settings, then Menu / Advanced, and disable "WiFi Optimization" if you have that item (see the attachment from Galaxy Nexus 4.2.2). According to Google Engineers, these "Optimizations" are not 100% compatible with all WiFi equipment, and it's a good test to try and disable them.
PS - not sure if K9 would report these errors at all.