If I upgrade from Lightroom 5.7 (which I bought and have a serial number for) to Lightroom 6, do I still have all my pictures?
Your pictures are not changed in any way when you upgrade.
When you will install Lightroom 6 and open your Lightroom catalog for the first time Lightroom will ask you to upgrade it.
Select yes and you will have all images,edits in the new version.See How to create and manage catalogs in Photoshop Lightroom for more info.
So nothing will be lost.
Hope that helps!!