Is there a particular word for it and what about the strings, are they just called strings?
The term used for the device used to control a marionette puppet (usually much more complex than just a simple cross) is, perhaps unsurprisingly, called a "marionette controller".
I found this through a quick Google search for "marionette puppet parts" which turned up images displaying the names for different components of a marionette puppet and its controller, and a followup search for "marionette controller" turned sites such as this shop that seem to confirm that name.
The cross in the picture is the control bar, and is used primarily with the kind of puppet in the picture, called a marionette
Marionettes, or "string puppets", are suspended and controlled by a number of strings, plus sometimes a central rod attached to a control bar held from above by the puppeteer.
As far as I'm aware, the strings are just called strings.