CustomCSS v Style sheet
...is probably more down to how you have done other customisations and your theme - if you are using a theme fairly much as it is from a download (or core) then i would recommend using the customcss setting, as any additional stylesheets could get lost in an upgrade. If it is largely your own custom theme and you already have customisations and maintenance then you could use a stylesheet. If you need portability between sites, stylesheets will transfer as part of the theme, customcss would be separate, but customcss on a single site will remain across an upgrade. So, that particular aspect is very much down to you and how you maintain your theme.
Recognise courseid as a selector
...Yes, it's a class in the <body>, along with many other very useful classes. If you use developer tools/firebug/view page source, then you will see it in the body tag.
Hope that helps