That's actually quite a lot of questions!
Let's start with the last one - No, you don't need to install another copy of Apache. In fact Yum/RPM won't let you. It is 'simply' a case of configuring Apache...
Of course, you don't actually need a second adapter for a second IP address - you can configure multiple addresses on a single adapter, known as Virttual IPs or VIPs.
And finally, you don't even need multiple IP addresses for multiple web sites - Host Headers can be set up in the web server to direct traffic for a particular URL to a specific set of content (as long as you are not using SSL). Or you can use a port other than the default, but this requires the user to specify the non-standard port, e.g. http://192.168.0.2:81
if the site is bound to port 81 rather than the default of 80 for HTTP.
But I digress - if you are running F16, click on the network icon on the Panel, select "Network Settings" from the menu (or select the System Settings applet under System Tools) and see whether it lists your new adapter and lets you configure it.