@tempest You can further reduce the values by manually editing parameters $120 (X) or $121 (Y) and probably also reduce maximum feed rate for the axis. I recommend you also use Dynamic Laser Power as with low acceleration the start and stop will have more burn without this feature.
Say you are using Y axis for the rotary, you would use debug mode and type these commands (examples):
$111=800
$121=20
@tempest T2Laser has a reduced acceleration option. You will need to calibrate the axis (to get the steps correct), this can be done in T2Laser but with a rotary you may need to manually adjust. Other Grbl settings can be changed using debug mode.
@wduensing No. The COM port would not show up in device manager if the Nano were bad. This sounds like a software issue, you can use my T2Laser free trial to test it (no need to change any firmware or settings, just connect to the COM port using default values).
@hami Correct. 1/16th microstepping is going to be smoothest but also the lowest torque. Full mode would have maximum torque but is very rough stepping. Resolution and feed rates are also considerations but this also varies depending on the drive mechanism (e.g. ballscrews improve performance but cost more and may need higher torque).
@péter-veizer If these were sheets of flat card I would say yes but it looks like corrugated cardboard and that is much more difficult. I would use my 40W CO2 for this but I am not saying it won't work with the diode (use a 5.5W for cutting and air assist).
@hami The default microstep setting is 1/16th, the Eleks board does not have switches to change this so you will need to remove pins on the A4988 or cut traces. MS1, 2 and 3 will all be HIGH for default setting so it's easy to change, if you want 1/8th microsteps just remove pin MS3 and ground it on the driver.
Do not change the dip switch under the Nano, it changes from internal power (5VDC from USB) to external power supply but since you are using 12VDC that will not end well 
for the Nano.
This is usually caused by backlash (a symptom of loose belts), circles are a good way to check for backlash as typically the ends will not connect if you have this issue. T2Laser includes an option to disable bi-directonal mode which is another good way to check if backlash is the problem, as when the laser always scans left to right it will effectively mask the issue.
I did not have any spare pieces but I don't identify this as part of the machine either.
https://www.dropbox.com/s/2ljv0huyyocg0rm/T2Laser_EleksDraw_Setup.pdf?dl=0
@Neil2 International Rectifier (IR) LR7843 or equivalent. It's a Power MOSFET in a D-Pak or TO-252AA package.
I wrote T2Laser software that supports Grbl 1.1 and has support, you will get much better performance from your machine.