The resistor limits the current into the base of the darlington pair... You could burn out the device without a limiting resistor.
The best suggestion is to follow the data sheet for the device.
Can't tell how 'big' the motor is, but generally you don't want high current loads going through the micro-controllers pcb... The trace lines are pretty small and you could damage them.. Run the motor ground from the Darlington (you have it marked -12V) to the 12V supply ground, not to the controllers pcb ground...
You will have to have a common ground between the darlington and the other devices.
To make it clear the only reason for a ground between the darlington and the micro is for the signal, very low current compared to the current the motor may draw.
Think about what's happening in this operation... The micro puts out 5V (or 3.3V depening on which micro) that drives the darlington. If the voltage on the input of the darlington is too high it will draw more current than is required. This damages the device.
The data sheet will tell you how much current the darlington needs to switch on and you compute your resistance based on that.
The darlington itself will allow current to flow to the motor... Too much current draw from the motor can damage/overheat the darlington.
There is really too much 'missing' information for me to really give you any numbers.
If it's a low current motor, the darlington may run it fine... if too much it will overheat I can't tell you from the limited information you have supplied...
Your 'cycle' rate is low 5 seconds on and 5 minutes off. That gives it time to cool down.. and is generally taken into consideration in the design stage. You can exceed the limits under certain conditions.
In short, you can damage your Arduino. It is similar to shorting your wall outlet where the fuse will blow if you try.
You do not need / want a resistor in the ground line.
Your topic has been moved to a more suitable location on the forum. Installation and Troubleshooting is not for advice (nor for problems) your project.
What will do that? The circuit? Edit - oh, I see now. Why are you reminding me of that? Is this a way of saying, "hey I don't have to heatsink it" or something like that?
Don't waste your time giving advice to @arpa123 , it won't be followed. They have already been advised to choose a MOSFET over a Darlington, and ignored that advice.