2011 MacBook (10.6.8) connects to AirPort but intermittently won't connect to webpages
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.