I don’t think this is possible, because there can always only be one window that receives input from mice and keyboards, i.e. one window that has the focus.
But you can run the program you want to automate in a separate X-server. So do
X :1 &
This way you can also use the whole screen. The downside to this approach is, that you still can not see your usual desktop and the controlled one at the same time. So either start the xdotool commands from the new X-server and leave them running in the background, or use a tool called
xpra. It allows you to run a new virtual X-server and connect to it from the default X-server. To set it up you run
xpra start :1
and to view the program run
from another terminal in the original X-server.