The issue is that you do not filter by content type and as a result it renders all content (regardless content type) as a card. Obviously you haven’t set the card display for all content types and thats why it renders like that.
You can fix it by adding a Content Type Filter and display only the content types that you want.
The specific cards are custom made cards for the HSE website. They were made specifically for this website and they are not maintained by the centralized infrastructure. The alternative offered by the infrastructure is the Related Card from CERN Components.
In Related Card the result will be rendered like the following image (without the colors and instead of the dummy image it will have the image you assigned for the node)
About displaying the name as a link to the mail: By default the title of the card is a link to the node. Role and phone number: You can map those fields to the yellow and green fields in the image above.
How to create a Card Grid?: Please read the documentation of CERN Display Formats My cards are horizontal and take the full width: Read my answer in the previous comment. It is because you display nodes of content types that do not have the card display.