How can I have Jenkins run a script on another server and watch over the result?
This is basic "Master - Slave" behavior in Jenkins - check the Distributed builds for that.
A) Install a Jenkins-slave on the remote machine.
B) Make this slave run "Tied Jobs" only.
C) Set step '4.' in your example to run on the remote Slave (or 'Node').
D) Step '3.' in your example should probably be a 'post-build-trigger' of step '2.'
(and not a step of its own).
The result of step '4.' in your example will be available to the master-machine.