You have (at least) two separate problems, not including your decision to stay with XP.
The "can't find a certificate" error typically means that your wifi network settings are misconfigured.
- Open Network Connections (Start > Run > ncpa.cpl > OK)
- Right-click on the icon for your wireless adapter and select Properties
- Click the "Wireless Networks" tab
- Ensure that there is a check in the box to "Use Windows to configure my wireless network settings." Note that in order to use Windows, you must ensure that any software that came with the wireless adapter is not also running.
- Assuming that your wireless network is listed under "Preferred networks," select it and click "Properties"
- On the "Association" tab, ensure that the authentication and encryption boxes are properly configured for whatever level of encryption you configured on your router (e.g., WPA2-PSK).
- Click the "Authentication" tab and clear the check box to "Enable IEEE 802.1x authentication for this network."
- OK your way out.
Alternatively, do steps 1 - 4 and when you get to step 5, select your wireless network and "Remove" it. Reboot. "View available wireless networks," find yours, click connect, and enter the password when prompted.
Windows Updates no longer works until you have service pack 3 installed. To overcome this problem you need to get to SP3 manually.
First read this article -- http://support.microsoft.com/kb/950717
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If you have n HP Pavilion, Presario or Media Center that has an AMD processor you need to install this patch before installing SP3 -->
http://h10025.www1.hp.com/ewfrf/wc/softwareDownloadIndex?cc=uk&lc=en&dlc=en&softwareitem=pv-60484-2
Download SP3 for XP from here -->
http://www.microsoft.com/download/en/details.aspx?displaylang=en&id=24
Ignore the text that says that it is for network installations and also ignore the text advising Windows/Microsoft Update if only updating one PC.
Logged in using an account with "Computer administrator" privileges, double-click on the saved file to install WinXP SP3. Follow all prompts; be patient, and reboot twice when the install completes.
Do Resolution Method 2 here -->
http://support.microsoft.com/kb/943144
Go to Windows Update to get the (many) post-SP3 critical updates.