You will not be able to use any robotic arm as a 3D printer for the simple reason that it is not a 3D printer. Accuracy aside, a 3D printer has a printing head with all supporting electronics and hardware to control the temperature and flowrate (how much plastic should be extruded). If you really want to print and not just waste time you buy a proper 3D printer. A cheap 3D printer will give much better results.
You would be able to make a 3D printer to draw and paint much faster and much more accurately than dobot by replacing the head with a pen holder. Just pick a 3D printer that is open-source, something like this would do http://www.gearbest.com/3d-printers-3d-printer-kits/pp_343643.html
You may be able to make dobot play chess though. Depending, of course, on your definition of "playing chess". Do you mean just repeat a certain combination of figure moves? If so, then yes, dobot can do that. If something more, then it would be quite difficult to control it.
Dobot can't draw with rigid drawing utensils, like a regular pen, because of the backlash. With soft utensils, like a brush, there is compensation for backlash in software, but it does this task extremely slowly, because of the poor choice of electronics and poor software design. https://github.com/maxosprojects/open-dobot is a little more flexible, but there is no compensation for backlash there yet (it is non-linear), so you won't get perfect results.
In this price range there is nothing that can execute all kinds of tasks accurately. Dobot is somewhat better than the rest in just a subset of tasks, but is worse in others.
Technically, a 3D printer can also be modified to play chess if you don't mind slow Z-axis movements. Delta printers don't have this limitation, but they are not as popular as they require a little faster control electronics due to the non-linear control, just like dobot needs (but doesn't have).
I might be missing be something, but the unassigned pins don't seem readily accessible - The dobot shield covers them. Are you suggesting I solder wires to the underside of the arduino board? That seems a little risky. I could see the wire touching adjacent pins. If there's some way to access these pins, I'm all ears.
Problem: As far as I can tell, there are no gpio pins readily accessible when using the dobot board. Furthermore, the I2C pins are being used by the dobot shield. There's lots of tasks I need to have the robot arm do that require different electronics, but to give an example, in one task I need to control 3 electronic devices: 1) a can stack stepper motor linear actuator 2) a differential pressure sensor that communicates via I2C 3) a solenoid. All are part of an electronic pipettor. Without the I2C pins, let alone GPIO pins, I can't do this. Limit switches and encoders are going to take up pins as well.
The timing problem is referencing the raspberry pi timing problem that you brought up. If I were to control the Dobot movement with the arduino/dobot shield from the pi, and use the GPIO pins on the raspberry pi for other peripherals (such as the can stack stepper motor I mentioned), I would still have a problem with the timing since I would be driving a stepper motor (the can stack stepper motor).
Perhaps to address all these issues I could connect another arduino serially to the pi, but I can barely program a serial protocol for one arduino, let alone 2. Instead of messing around with serial communication, one of my questions was is there some way I can avoid serial communication altogether? For example, the beaglebone board seems to have something called a PRU (programmable real-time unit, http://beagleboard.org/pru), which addresses the timing issue that the pi has. From what I can tell, the beaglebone is similar to having a microcontroller and linux computer on one board.