I had a nice long post ready to go, but after re-reading the original, it had me a little lost. You use the term
Client #1 (our HTPC).
It appears this is the Media Server? If so, let's call the Server and not Client.
If this is correct, then yes, any changes done here will show up on the other 2 clients #2 and #3
To get the changes on Client #2 and #3 back to Server, you need to authenticate from client to server to sync your changes.
Here are the steps to set this up.
On Server (HTPC), go to Tools - Options - Media Network.
Check "Use Media Network to share the library and enable DLNA". Also check "Authentication (must be on to use Library Server Sync)
This will pop up a window asking for a username and password. Input your choosing here.
On Client PC, connect to library. Connecting to library will ask for the username and password you entered on server pc.
Then go to Tools - Options - Media Network. On right pane, click Client Options and check Automatically sync changes made on client back to server.
Do this to both clients.
After making changes or playlists, you can do a manual sync by going to File - Library - Sync changes with Library Server. Any changes will also sync when you close MC on Client PC.
Hope this helps