Regarding using a listing image as a marker that's not possible. You can use custom markers, but you cannot dynamically create markers using images from the posts.
You can add whatever markup and CSS you like to the filter section of a View to customise it to your needs.
Views does not offer any way to customise the HTML markup of individual filters, but you can split some filters into one tab and other filters into another. And in the Custom Search settings you can choose "Show only available options for each input" to hide options that are not possible when another filter option has been selected.
The problem I see trying to reproduce what appears on the page you linked to is that the first tab has Buy or Rent options, and when you choose one the filters available in the second tab change accordingly (e.g. rent vs purchase price).
They are effectively two separate Views, one to display rental properties (one CPT) and another to display for sale properties (another CPT).
You can't switch the Views on the front-end in the same way as on the example site, you need separate searches to search for separate things, but for any one search you can design the filter output section so that the filters are spread across tabs.
