No, this is not possible: the only "commands" you can send over SSH (and HTTPS) are
hg push and
They're not really sent as "commands", instead Mercurial uses it's own protocol which allows a client to query a remote server for a few things like "do you have this changeset?", "what are the branch heads?", and "please accept this bundle!".
When you use SSH, your Mercurial client will make a SSH tunnel and start
hg serve --stdio on the remote host. That
hg serve command is what speaks the Mercurial wire protocol I talk about above. Hosting environments such as Bitbucket are locked down so that this is the only command you can execute over SSH — they don't want you to execute arbitrary commands on their servers!
As for managing a remote repository on your own webserver: You need to login with SSH and create an empty repository on your Unix server:
$ hg init my-website
Then you can push to the server from your local machine:
$ hg push ssh://your-server.com/path/from/home-dir/to/my-website
Since you're talking about managing a website, you might want to add
changegroup = hg update
.hg/hgrc file in the remote repository: it will make Mercurial run
hg update after every push into that repository. This means that the working copy will be updated with the latest files and so your webserver can serve these to the world.