I cannot access the site you linked as a Server Security Applet asks Login Name and Password.
I enabled a Private Reply for you to provide them.
Toolset requires knowledge of HTML and CSS/JS, as we do not provide Support or GUI to add lots of CSS or HTML, it is more about building the structures, and then apply custom HTML, CSS or JS either manually in the editors, or for example with some page builder like Beaver Builder or similar.
This cannot be done in the Search HTML section - there you need to apply your own manually created styles and HTML.
If you plan to modify the "native" inputs I suggest to always choose "legacy" or "raw" Styling output when you insert your filters, and not the Bootstrap Fully styled output, as that will make it easier to add your own styles.
Background colors on inputs that can have a checked value can be added with JS for example using the addClass() and removeClass() jQuery functions depending on the value that the inputs have.
CSS itself even supports :hover, :checked, and more.
There are some hints on here as well:
https://stackoverflow.com/questions/34388696/how-to-change-the-background-color-on-a-input-checkbox-with-css
https://stackoverflow.com/questions/24135355/uncheck-a-checkbox-using-css
versteckter Link
versteckter Link
This is not something we can widely assist with as it's a design task, but I think with above details you should already be able to proceed!
Related to "When you select "Rechtsanwälte" the first displayed item looks other than the other. This only happens at this selection with this item. Can you please tell me, why this happens only to this item on this selection?", I am not sure, but this is not due to Toolset native functions.
Instead, it is due to any style or rule applied that does this whenever that value is selected.
Toolset has no "automated" methods to change appearances on the fly.
Does that happen as well when you remove all CSS and JS from those Views?
The posts then should display as a simple unformatted output.
If that is not the case it means there is more HTML or CSS in the Loop of the View, that may produce the issue.
Let me know if something is unclear (private reply is as well active)