I now would like to modify the style, especially the spacing (some examples above in the image - these are just examples to set the idea). The closest I found in the documentation is Styling - custom:button-card documentation but it is still too much of a high level.
Is there a place I should look into? Or a naming condition for elements or classes, I could address in the styling? Or should I dig into the code?
The link you pointed to is a good start. Next is to check out the different layout options. The image you show looks to have the icon layout, but no icon. You can either use one of the standard layouts or define your own. Layouts are grid based so you can first look to adjust areas by adjusting the grid styling with usual grid css as config items in grid style. Once you have mastered this, then you might need further style tweaks. But don’t jump to css height, padding, margins to alter layout without understanding and mastering the inbuilt layout options.
As for overall size and fit with other cards, if you use sections make sure you have the right options set. If you use GUI to create a button-card, valid defaults are set depending if you are in sections dashboard or say masonry.
I hope this helps you on the way with button-card. If you have any further questions happy to help.