Lightroom has an "organizer" called the Library Module. OPINION: I prefer the Lightroom Library Module to the PSE Organizer.
You cannot (easily) use Lightroom as a photo editor with Photoshop Elements. There really isn't a good way to do this. You can work in the opposite direction easily, organize in Lightroom and edit in Lightroom and then do additional editing in the PSE Editor.
So, probably the best combined usage of these two is to Import your PSE catalog into Lightroom (not import the images, import the catalog, despite what the title at that link says) and then not use the PSE Organizer any more. Use Lightroom for editing and PSE Editor when you need more editing tools and more editing power.
Yes, both softwares run on Windows 7 (Lightroom 6 and later require the 64 bit version of Windows 7)
As Paige says if you import as a Catalog Lightroom will automatically make links to your image files. Also if you have added keywords they will be recognized in Lightroom. Any Organizer Albums that you have will be imported as Lightroom collections.