You can start by using the Collect Files function to generate a report and then perhaps you can find a bunch of scripts on AEScripts.com. There's definitely stuff similar to Collect Files, but counting shape groups and other finer granularity levels may be something you have to add if you realyl care about the statistics. That's stuff nobody usually even considers relevant, given that they are parametrically generated internal entities.
File > Dependencies > Collect Files > Generate Report Only will write a text document that lists the Compositions and counts external source footage. It does not; however, not count Layer nor Layer Source types that After Effects generates from within the project file.
For something customized, a custom script should be able to do it. I hired a someone to write a custom script that gets Comp names and rendered filenames in one click. The time it saves me to do a Collect Files - Generate Report Only and always set the Render Queue to write a log file was well worth it.