This is indeed quite broken in Mountain Lion. Once you've fixed up the default route as you describe in the question, you're still left with the problem that Mountain Lion is giving its bridge interface address to clients as both the router address (which is correct) and as the DNS server address (which isn't).
Verify that this is the problem by entering an HTTP server IP address into the address bar on a client web browser when connected through your Mac after you fix default routes, and it should load up fine.
My solution to this problem is to fix up the route as you describe -- which could be automated, of course -- and to keep BIND (aka /usr/sbin/named) running in the background on my Mac in a forward-only configuration, forwarding all queries to Google's public DNS servers. This doesn't fix the underlying brokenness in Mountain Lion, but it makes things start working for the clients.
A couple useful resources:
http://www.macshadows.com/kb/index.php?title=How_To:_Enable_BIND_-_Mac_OS_X%27s_Built-in_DNS_Server (how to fire up BIND on OS X)
http://gleamynode.net/articles/2267/ (how to configure BIND for forward-only operation -- of course you will not want to make BIND only listen on 127.0.0.1)
It would be far preferable for Apple to make this feature of their OS work as advertised, but in the meantime I've found this is a viable workaround.
The main thing that looks odd from the setup that you've mentioned above is the DNS server. It appears that the IP address your computer has allocated is being assigned as the DNS server? If you're not running DNS services on your computer, this won't work very well.
Try to remove the single DNS server (i.e., your computer's IP address) from the list of DNS servers, and see if the machine populates that list with other entries (that were potentially provided via DHCP). If there are no DNS servers listed in that dialog box after you remove the single entry, I'd suspect configuration problems on your network.
Alternately, it may help to manually configure that network interface to use Google's open DNS servers (8.8.8.8, 8.8.4.4) to see if they will respond correctly to DNS lookups.
You can use the Network Utility program at /Applications/Utilities to perform further network troubleshooting. For example, it should be possible to "ping" the DNS server, router (gateway) and some outside host from where you are. This lets you know that your machine can communicate with those devices. First, I'd recommend trying to use Network Utility to ping the DNS servers that are being provided via DHCP. Under the "Ping" tab, you can enter the IP address of each of those hosts, and click "Ping" to see if you get a response. If all of the DNS servers are responding to pings, see if you can ping the gateway (10.58.204.1). If you receive responses for pings to all of the 'internal' hosts, see if you can ping some outside host (gmail.com?) to see if your computer's network traffic is being routed to the Internet.
If all of the ping tests are passing, I'd recommend trying to perform DNS lookups on the servers that are showing up in the DNS table in Network System Preferences. You can use the Network Utility program to perform lookups, but it may be more helpful to diagnose the DNS servers individually using 'nslookup' through the Terminal program. Open Terminal from /Applications/Utilities. When the program opens, you'll see a command prompt. Type in the following queries, and observe whether or not you receive valid responses from all of the DNS servers provided:
nslookup gmail.com 172.16.2.5
nslookup gmail.com 172.18.82.11
nslookup gmail.com 4.2.2.2
Those servers should respond with some answer (and IP address for gmail.com) within milliseconds. If you're seeing the commands hang for excessively long periods, that particular DNS server may not be responding correctly.
It's odd that you're having intermittent issues when using Google's DNS servers...If you're on some larger private network (as it appears you are based on the private addresses being provided), it's possible that traffic is being filtered.
Lastly, the Awarenet profile that you're using is simply utilized for authentication to a wireless access point named "Awarenet" that uses 802.1x (WPA Enterprise) security to authenticate users (you're signing in as egoodwin). If you no longer use/join a Wi-Fi network named "Awarenet" (for work, or school?), the profile can likely be deleted.
Best Answer
Your wireless device appears to be in promiscuous mode. Maybe you are doing a packet dump or sniffing packets? If so, no worries. If not, I would investigate this further, as it sometimes yet not always indicates a compromised system being used as a stepping stone in further network penetration.
First, a quick and dirty network device refresh can be effectuated by creating a new location in the System Preferences, under the network portion. Look at the top, and under location, click edit locations. Then click the plus button, and give the new location a name.
Then please select your home network, and re-enter the password.
Your problem might have been resolved. If not, then please go back to the terminal and provide the output of these:
Hope that this helps.
F.