Can you remove a file from Mercurial without removing it from the filesystem like Git

For files you might want to change your command line slightly to the suggested command-line on the git-rm page

git ls-files -z | xargs -0 rm -f

This is much safer with more complex file paths. For directories you could try a similar strategy:

git ls-tree --name-only -d -r -z HEAD | sort -rz | xargs -0 rmdir 

It does however depend on how you would like to treat directories that contain files (often gitignored) that are untracked. The command line above should leave these files and their directories.

But it would be easy to change this to delete the directory, whatever the contents.

You seem to be looking for the git status command.

From man git-status

Displays paths that have differences between the index file and the current HEAD commit, paths that have differences between the working tree and the index file, and paths in the working tree that are not tracked by git (and are not ignored by gitignore(5)). The first are what you would commit by running git commit; the second and third are what you could commit by running git add before running git commit.