Mint should have selected the wl
kernel module for your Mini's wifi and it should be working immediately on install. No hassles. Assuming you don't have something funny happening with your router.
A few things for me to ask up front:
1. Before you installed Mint, your Mini came with either Ubuntu Hardy (lpia) or Windows. Did you toggle the radio off? From Hardy, did you use the aircraft-manager app to disable your wifi radio, or from Windows did you toggle the radio off? The BCM4312 has an internal firmware OFF switch that you may have toggled. Mint Gloria does not include the aircraft-manager to toggle this back on.
2. When you R-Click on nm-applet on the panel, is the checkbox CHECKED for wireless networking?
3. When you do "lspci -vvnn" and browse for your BCM4312, is the kernel module in use called "wl"?
4. Do you have an ethernet cable plugged into the Mini? Unplug it.
Based on this, I think your wifi is good, your card is enabled, you're using the right kernel module, and everything should work:
eth1 IEEE 802.11 Nickname:""
Access Point: Not-Associated
Link Quality:5 Signal level:199 Noise level:169
Rx invalid nwid:0 invalid crypt:0 invalid misc:0
The wl kernel module with the BCM4312 should configure the wifi as eth1, which is going to look odd to most of the Mint geeks here expecting wlan0 or something else. But eth1 is correct for a Mini-9. I think the whole problem is that you're simply not properly connected to your AP.
Go to nm-applet, R-click, edit your connections, and DELETE your wifi connection. Wait a few seconds, L-click on nm-applet, and reconnect to your AP. This reconnects as if for the first time.
What I don't understand is why your eth1 is identifying as addr:10.42.43.1 if your router is addr:192.168.1.254 and addr:192.168.1.33 is your desktop. Its like your wifi is connected to a different network entirely, not your own.