I'm able to use stylelint for .css files according to their usage guide. Is there any way to use it for inline and internal stylesheets? I could write code to collect all those by parsing my HTML documents... is that my only option? Do I have to use another CSS linting tool like CSSLint?
Yes, use the stylelint processor stylelint-processor-html
See also http://stylelint.io/user-guide/processors/ for a list of other stylelint processors.