Almost, the following will do what you want except that you'll also need one body style per level (the same as you need one heading style per level), instead of one style overall. And please note that I'm doing these in Word 2003 so they actual motions may be slightly different.
Build Styles
Get up the styles pane, IIRC in Word 2007 you hit the small down arrow on the bottom-right of the styles selector on the ribbon. Click on the New Style button, on Word 2007 I think this is a pictoral button of a letter A with a gleam/star on it.
Each new style should be Based On a corresponding Heading entry, this ensures they will stay within the correct numbering pattern, because they're using exactly the same pattern. Obviously, you'll need to remove bold, adjust the font size, adjust the paragraph spacing, and whatever else you need to ensure the body text appears how you want it to.
You probably will want to create BodyText1 based on Heading2, BodyText2 based on Heading3, and so on for however many levels you will need.
Instead, you might be able to base BodyText2 (etc) on BodyText1, and just adjust the numbering level, but for some reason this caused me a problem. Not sure why, but you may wish to try so you don't have to redefine the font size/etc for each BodyText style.
Adjust Table of Contents
Because the new styles are based on the Heading ones they will appear in the table of contents. Obviously, this is not desirable.
This is where my memory of Word 2007 memory becomes hazy, the following is purely Word 2003 - if it doesn't match, let me know and I'll update for Word 2007 once I'm home and can check directly.
Right click on your current Table of Contents and select Edit field..., then click the Table of Contents button to get to the TOC window. (You can probably get here when creating a brand new TOC as well)
Click the Options button (bottom-right) and then find each BodyText style in the list; clear the TOC Level field from each of these. This ensures the BodyText items will not appear in the TOC at all. Close each window until you get back to your document, you might be asked if you want to replace the current table, if so click Yes.
Use the styles
Now just ensure you set the relevant style on each paragraph as you write.
Best Answer
Assuming your page numbers are on a footer (or header), it seems that Word makes them "same as above by" default. Thus numbering goes across sections.
Activate your footer, look if there is a tab "same as above" on it. There is a button on a ribbon which removes that. After that your fields on a footer will be independent.