"Is there a css code to change the menu text color only on the homepage including the mouseover and active menu point?"
This can be achieved with help from the following link:
https://laytheme.com/documentation.html#css-based-on-current-page
"Can I change something so it adjusts the size automatically so that there are no blank spaces between the pictures?"
You should be able to choose "Fill Slides" in the Carousel settings?
Best wishes Barbara & talk soon 🌝 Richard