I'm a business user and trying to find out if Sharepoint is the right tool to manage our files. Our files consists of excel spreadsheets and Word documents. They include policies, procedures, and tools (mostly excel spreadsheets) that are mostly used for day to day operations. Thanks.
In my experience SharePoint is useful for Document Management for large companies because of its metadata tagging abilities. If you don't have thousands of files with different permissions required for them all, then I'd say it's probably better to just go with a fileshare. SharePoint takes a lot of resources to run and can be a pain to maintain.
Some other useful features of SharePoint are coauthoring, which allows more than one person to edit a document at once, the versioning as mentioned by DrFeelgood, and it's general ability to integrate with other Microsoft software like Lync, Outlook, Office.
I would think so, especially 2013. Your biggest benefit here will be versioning. If someone messes up a version of an important document you can retrieve older copies. Another great thing about 2013 is that they have SkyDrive Pro, which is essentially a site collection for yourself that you can map in Windows so you can use Windows Explorer with it.
If you want to get a bit IT-heavy, I think the coolest thing you could possibly do is create an app for a document that you have stored inside a list that has an app attached to it.
But just don't use SharePoint for document management, it's much more powerful.