@tomaja: there are no conflicts in CSS. It’s all about what rule takes precedence. When possible, do not use !important in your CSS code, especially when giving advice. And, if you don’t find any other way, at least do it properly: (there should be a space before the exclamation mark and the CSS declaration you’re making important should be valid). background-color: none; is not a valid CSS declaration.
Since you modified customizr and were able to create my-style.css I assumed you would replace the declaration on lines 28 to 30 in that file with my code. If you do it from custom CSS panel, you need to put .carousel-inner in front of my code, like this: