1. How do you display an online version of the form in the SharePoint 2007 site?
Once Forms has completed the render it will return a PDF (along with some other stuff). You can then take that PDF and write it to the appropriate SharePoint directory. I'm not a SharePoint expert, but I believe that it does have a WebDav interface.
You could always do both the render and the file write in a single LiveCycle workflow.
2. Once the user fills the form online and submits the data how do you capture the data?
When you make the call to Forms to render the form one of the parameters is the target URL. This is the location that your PDF will submit to using an HTTP Post. You'll need to create some code for the target (an ASPx or a Servlet) which will then call Forms to extract the data (using processFormsSubmission).
Since you are using SharePoint with LiveCycle, you may want to look at the LiveCycle SharePoint connector. Its in beta development now, but you can get at it by signing up with the Adobe pre-release site: prerelease.adobe.com
The connector may do a lot of what you are looking for
How did you go with the "LiveCycle ES2 Connector for Microsoft SharePoint Server 2007 and 2010 (Version 188.8.131.52)" ?
I'm interested in getting my Adobe Livecycle ES 2.5 (Suse Linux - JBoss) to work together with my SharePoint 2010.