Alright. So now we know that when you login as a regular user account and then su - root to launch firefox you are able to browse the sites.

Appears to be permission or browser setting issue but can't say for sure. We need to perform further testing. Run the following commands with test account from terminal:

Once you are in nslookup run the following:

after that

www.google.com
www.yahoo.com
www.hotmail.com

If you are able to get response from these sites then we are good to move on to next test.

Exit from nslookup by typing "exit" without quotes.

Run the following commands:

If you get ping response then we are sure that there is nothing being blocked when you are logging in as regular user. It has something to do with permission or browser config.

Once the above tests are done run the following commands and let us know the output:

1.2.3.In the last command it is firefox-3.6 because I am running that version. Could be different for you so type that accordingly.

# nslookup
> set debug
> www.google.com
Server: 199.242.236.1
Address: 199.242.236.1#53

------------
QUESTIONS:
www.google.com, type = A, class = IN
ANSWERS:
-> www.google.com
canonical name = www.l.google.com.
-> www.l.google.com
internet address = 74.125.227.48
-> www.l.google.com
internet address = 74.125.227.49
-> www.l.google.com
internet address = 74.125.227.50
-> www.l.google.com
internet address = 74.125.227.51
-> www.l.google.com
internet address = 74.125.227.52
AUTHORITY RECORDS:
-> google.com
nameserver = ns4.google.com.
-> google.com
nameserver = ns1.google.com.
-> google.com
nameserver = ns2.google.com.
-> google.com
nameserver = ns3.google.com.
ADDITIONAL RECORDS:
-> ns1.google.com
internet address = 216.239.32.10
-> ns2.google.com
internet address = 216.239.34.10
-> ns3.google.com
internet address = 216.239.36.10
-> ns4.google.com
internet address = 216.239.38.10
------------
Non-authoritative answer:
www.google.com canonical name = www.l.google.com.
Name: www.l.google.com
Address: 74.125.227.48
Name: www.l.google.com
Address: 74.125.227.49
Name: www.l.google.com
Address: 74.125.227.50
Name: www.l.google.com
Address: 74.125.227.51
Name: www.l.google.com
Address: 74.125.227.52
> www.yahoo.com
Server: 199.242.236.1
Address: 199.242.236.1#53

------------
QUESTIONS:
www.yahoo.com, type = A, class = IN
ANSWERS:
-> www.yahoo.com
canonical name = fp3.wg1.b.yahoo.com.
-> fp3.wg1.b.yahoo.com
canonical name = any-fp3-lfb.wa1.b.yahoo.com.
-> any-fp3-lfb.wa1.b.yahoo.com
canonical name = any-fp3-real.wa1.b.yahoo.com.
-> any-fp3-real.wa1.b.yahoo.com
internet address = 209.191.122.70
AUTHORITY RECORDS:
-> wa1.b.yahoo.com
nameserver = yf1.yahoo.com.
-> wa1.b.yahoo.com
nameserver = yf2.yahoo.com.
ADDITIONAL RECORDS:
-> yf1.yahoo.com
internet address = 68.142.254.15
-> yf2.yahoo.com
internet address = 68.180.130.15
------------
Non-authoritative answer:
www.yahoo.com canonical name = fp3.wg1.b.yahoo.com.
fp3.wg1.b.yahoo.com canonical name = any-fp3-lfb.wa1.b.yahoo.com.
any-fp3-lfb.wa1.b.yahoo.com canonical name = any-fp3-real.wa1.b.yahoo.com.
Name: any-fp3-real.wa1.b.yahoo.com
Address: 209.191.122.70
> www.hotmail.com
Server: 199.242.236.1
Address: 199.242.236.1#53

------------
QUESTIONS:
www.hotmail.com, type = A, class = IN
ANSWERS:
-> www.hotmail.com
canonical name = dispatch.kahuna.glbdns.microsoft.com.
-> dispatch.kahuna.glbdns.microsoft.com
internet address = 65.55.72.119
-> dispatch.kahuna.glbdns.microsoft.com
internet address = 65.55.72.135
AUTHORITY RECORDS:
-> glbdns.microsoft.com
nameserver = glb1.glbdns.microsoft.com.
-> glbdns.microsoft.com
nameserver = glb2.glbdns.microsoft.com.
ADDITIONAL RECORDS:
-> glb1.glbdns.microsoft.com
internet address = 204.79.195.7
-> glb2.glbdns.microsoft.com
internet address = 65.55.117.2
------------
Non-authoritative answer:
www.hotmail.com canonical name = dispatch.kahuna.glbdns.microsoft.com.
Name: dispatch.kahuna.glbdns.microsoft.com
Address: 65.55.72.119
Name: dispatch.kahuna.glbdns.microsoft.com
Address: 65.55.72.135
>

I think this means I can't get response from these websites, right?

I can't get ping response either.

No. That means that nslookup is able to resolve the name of those websites. I can see that you run nslookup with root user. You have to run that with test user account. What was the result of the other tests that I have suggested?

What you mean by you cannot get ping response. Paste the output.

[test@swan ~]$ nslookup
> set debug
> www.google.com
Server: 199.242.236.1
Address: 199.242.236.1#53

------------
QUESTIONS:
www.google.com, type = A, class = IN
ANSWERS:
-> www.google.com
canonical name = www.l.google.com.
-> www.l.google.com
internet address = 74.125.227.52
-> www.l.google.com
internet address = 74.125.227.48
-> www.l.google.com
internet address = 74.125.227.49
-> www.l.google.com
internet address = 74.125.227.50
-> www.l.google.com
internet address = 74.125.227.51
AUTHORITY RECORDS:
-> google.com
nameserver = ns4.google.com.
-> google.com
nameserver = ns1.google.com.
-> google.com
nameserver = ns2.google.com.
-> google.com
nameserver = ns3.google.com.
ADDITIONAL RECORDS:
-> ns1.google.com
internet address = 216.239.32.10
-> ns2.google.com
internet address = 216.239.34.10
-> ns3.google.com
internet address = 216.239.36.10
-> ns4.google.com
internet address = 216.239.38.10
------------
Non-authoritative answer:
www.google.com canonical name = www.l.google.com.
Name: www.l.google.com
Address: 74.125.227.52
Name: www.l.google.com
Address: 74.125.227.48
Name: www.l.google.com
Address: 74.125.227.49
Name: www.l.google.com
Address: 74.125.227.50
Name: www.l.google.com
Address: 74.125.227.51
> www.yahoo.com
Server: 199.242.236.1
Address: 199.242.236.1#53

------------
QUESTIONS:
www.yahoo.com, type = A, class = IN
ANSWERS:
-> www.yahoo.com
canonical name = fp3.wg1.b.yahoo.com.
-> fp3.wg1.b.yahoo.com
canonical name = any-fp3-lfb.wa1.b.yahoo.com.
-> any-fp3-lfb.wa1.b.yahoo.com
canonical name = any-fp3-real.wa1.b.yahoo.com.
-> any-fp3-real.wa1.b.yahoo.com
internet address = 209.191.122.70
AUTHORITY RECORDS:
-> wa1.b.yahoo.com
nameserver = yf1.yahoo.com.
-> wa1.b.yahoo.com
nameserver = yf2.yahoo.com.
ADDITIONAL RECORDS:
-> yf1.yahoo.com
internet address = 68.142.254.15
-> yf2.yahoo.com
internet address = 68.180.130.15
------------
Non-authoritative answer:
www.yahoo.com canonical name = fp3.wg1.b.yahoo.com.
fp3.wg1.b.yahoo.com canonical name = any-fp3-lfb.wa1.b.yahoo.com.
any-fp3-lfb.wa1.b.yahoo.com canonical name = any-fp3-real.wa1.b.yahoo.com.
Name: any-fp3-real.wa1.b.yahoo.com
Address: 209.191.122.70
> www.hotmail.com
Server: 199.242.236.1
Address: 199.242.236.1#53

------------
QUESTIONS:
www.hotmail.com, type = A, class = IN
ANSWERS:
-> www.hotmail.com
canonical name = dispatch.kahuna.glbdns.microsoft.com.
-> dispatch.kahuna.glbdns.microsoft.com
internet address = 65.55.72.39
-> dispatch.kahuna.glbdns.microsoft.com
internet address = 65.55.72.23
AUTHORITY RECORDS:
-> glbdns.microsoft.com
nameserver = glb2.glbdns.microsoft.com.
-> glbdns.microsoft.com
nameserver = glb1.glbdns.microsoft.com.
ADDITIONAL RECORDS:
-> glb1.glbdns.microsoft.com
internet address = 204.79.195.7
-> glb2.glbdns.microsoft.com
internet address = 65.55.117.2
------------
Non-authoritative answer:
www.hotmail.com canonical name = dispatch.kahuna.glbdns.microsoft.com.
Name: dispatch.kahuna.glbdns.microsoft.com
Address: 65.55.72.39
Name: dispatch.kahuna.glbdns.microsoft.com
Address: 65.55.72.23
> exit

[test@swan ~]$ ping www.google.com
PING www.l.google.com (74.125.227.51) 56(84) bytes of data.

--- www.l.google.com ping statistics ---
12 packets transmitted, 0 received, 100% packet loss, time 11000ms

[test@swan ~]$ ping yahoo.com
PING yahoo.com (72.30.2.43) 56(84) bytes of data.

--- yahoo.com ping statistics ---
6 packets transmitted, 0 received, 100% packet loss, time 4999ms

[test@swan ~]$ ping hotmail.com
PING hotmail.com (65.55.72.183) 56(84) bytes of data.

--- hotmail.com ping statistics ---
2 packets transmitted, 0 received, 100% packet loss, time 999ms

[test@swan ~]$ ls -a /home/test/.mozilla/firefox
. .. dcv50nks.default profiles.ini
[test@swan ~]$ ls -l /usr/bin/firefox
-rwxr-xr-x 1 root root 4526 Dec 9 2008 /usr/bin/firefox
[test@swan ~]$ ls -l /usr/lib/firefox-3.0.5/firefox
-rwxr-xr-x 1 root root 47540 Dec 9 2008 /usr/lib/firefox-3.0.5/firefox

I pinged, but there is no response.
So I use Ctrl+C to stop it and pasted the result above.
Is there anything wrong?

Alright. Lets sum it up:

1. Regular user cannot ping websites but is able to get response via nslookup (Try it from root if the same behavior then ICMP is blocked somewhere)
2. If root user can ping websites and is able to get response via nslookup (Then regular user is either blocked by iptables or could be something else preventing access to the internet)
3. After going through all outputs and the comments you made on proxy setting (you said it is same for both test and root user) there is nothing appears to be wrong on browser part.
4. su - root test we performed in the terminal shows that there is something (could be iptables or something else) that is only allowing root user to pass through this system.

Do you have any other system in the same network. Test from that system if you are able to surf internet using normal user. If yes, then there is something on this system which is blocking internet access for normal user. If not, then there is something centralized which is blocking internet access for normal user and only allowing internet access on the basis of user name such as root.

I am logging off for now but this was my observation so far. Perform the test that I have suggested in this post (if possible) to see if there is something centralized that is blocking or it is this system which is blocking.

root user can't ping websites but is able to get response via nslookup

it's the same behavior in the root account.

As I said that if you get the same response for ping even for the root account then your firewall is blocking ICMP requests. As you are able to browse sites via root account I am pretty sure that it is your firewall that is configured that way or there is a proxy server which is setup that way.

Thank you!