I hope your planning on going beyond what Carto does. Baremaps currently doesn't render highway=busway either, leading to awkward gaps in the map. It's an accepted tag supported in every renderer except Carto (look at OsmAnd or Organic Maps for instance), because the two or three active maintainers remaining (having lost a number of maintainers in the past two years) don't like the tag. Carto has pretty much stagnated despite its past achievements.

As the default renderer for openstreetmap.org, Carto is currently an embarrassment for OpenStreetMap.

Thank you for describing this issue, I created an issue on GitHub to track it. The fact that OpenStreetMap Carto is used in the OpenStreetMap documentation greatly accelerated our styling process (e.g. https://wiki.openstreetmap.org/wiki/Map_features). That being said, it is a loose port, and I hope we will soon be able to provide other derivatives, such as a positron style.

Are coordinates stored in Web Mercator?

Yes, as the tiles are created directly from Postgis, reprojecting the coordinates would take too much time.