The mechanical movement of the printhead is capable of going down to 0.01mm. This is accomplished by the accurate stepper motors that allow 100 steps for every mm of movement in the X and Y directions. The final print accuracies are going to be dependent upon a few factors such as what material you are using, and the diameter of the nozzle. Various filaments will expand and retract at different rates due to each specific thermal properties and die swell. Luckily, all results are very repeatable. If due to thermal shrink and expansion you are off by 0.5% in the X direction, you can scale your model by 0.5% in X to increase accuracy. Straight out of the box you can expect accuracies within 0.25mm of intended, and with some fine tuning of the model you can improve upon that accuracy.
In regards to your resolution issues, some photos or description would help to narrow down what might be causing them. Are you getting stringing? Banding? Warping?