When you say "unknown sources is checked off," you mean that the checkbox is actually checked, right? That means Android will allow you to install apk's from unknown sources.
You also say that when you're done downloading, a popup says "Can't open file." Does this happen without doing anything in between? Apk files typically shouldn't open automatically if you're downloading them directly from the web. It should download into your Downloads folder, then you navigate to that folder and tap the apk file to run the package installer.
Downloading apk files from the web can be somewhat risky, so keep in mind that you might be downloading malicious files as well. And let's give you the benefit of the doubt that you're not downloading pirated apps, because we don't discuss that here.