The first block of commands you report as non-working only operate if the rundown has remote control enabled. The enable/disable of the commands is via the Rundown Menu “Allow Remote Triggering” item. The tab at the top of the rundown page has a red lightning fash to show it is remote enabled.
Please note the /control/pause does not work in the distributed app versions. If you are comfortable with compiling applications you can fix the issue. The documented fix can be seen https://github.com/CasparCG/client/issues/304. The required tools to create the updated QT app are all available at zero cost.
There is a further control mechanism that you can use. Create a new rundown and add multiple instances of the Playout Command from the tools library. Edit the instaces to have a named UID (eg c1c2 etc). Set the playout command to the required action, so UID C1 issues a play command, UID C2 issues a stop command etc. These actions trigger even when the rundown is not “focused”, and operate on the selected item in the target rundown that has focus. You still need to enable remote control on the main rundown so that you can send the /control/up or /control/down messages. The bad news is that the Playout Command pause/resume do not work unless you edit the client source code and re-compile. The required edit is documented at https://github.com/CasparCG/client/issues/301.