Custom icon on coordinates

Of course it is possible.

[leaflet-map fitbounds]
[leaflet-marker lat=... lng=... iconurl="https://my-domain.tld/path/to/myicon.png"]My Marker[/leaflet-marker]

Maybe you need to adjust some options. I can help you, if you give me an example.

1) Is it possible to get another design for the card?

Here can you see many possible maps. You need to define one in the Leaflet Map settings. If you want to switch between the various designs, take a look to Extensions for Leaflet Map and some examples.

2) For the marker text, the tooltip is very far from the marker: https://snipboard.io/IFTbgX.jpg

You need to adjust the popupanchor. Please give me an url of the marker, then I can help you more.