Some ideas of things to try:
Do you have any other wireless clients you can use at the same time, to see if all devices drop off the network at the same time?
Does the AirPort Menu Extra switch to showing no bars when the dropoffs happen?
Try pinging to your router's private IP address and leaving the ping going in the background all the time, and see if the pings start failing when the problem occurs. I ask because I've seen too many people blame "Wi-Fi" for connectivity loss when actually the Wi-Fi link stayed up fine, it was their DSL line that hiccuped.
Run this command:
sudo /usr/libexec/airportd debug +AllUserland +AllDriver +AllVendor
...and then reproduce the problem and share your /var/log/kernel.log and /var/log/system.log from the time the problem occcured. (Note: that command does not fully persist across reboots, so you'll need to reissue it after rebooting.)
Update: The multiple "MAC Auth Succeeded" messages for the same AP in rapid succession imply something weird is going on. It would be useful to see an 802.11 monitor-mode packet capture of what's going on. It's best to do the capture on another machine, otherwise you won't see all the packets you need.
If doing the capture from another AirPort-equipped Mac, the simple way is to run this command:
sudo /usr/libexec/airportd en1 sniff 1
(...assuming your AirPort card is
en1, and your AP is on channel
1; otherwise, tweak those arguments to the command.)
While the command is running, reproduce the problem, then Ctrl-C out of the command, and post the packet capture. Warning: Make sure you're not sending any unencrypted confidential information on an unsecured 802.11 network on that channel at the time you're running that capture, otherwise those details will be captured.
If you don't have another Mac to do the trace from, you can run this command on the machine that's having the problem:
sudo tcpdump -i en1 -I -y IEEE802_11_RADIO -s 0 -w capture.pcap
...then reproduce the problem, then Ctrl-C, etc.