How does the position of the gound influences the node voltages?
The currents are all the same through the components. The relative voltages between each node are the same. The difference is the absolute voltages are not the same. No matter where you place the ground, the nodes will have the same voltages relative to one another. The ground is a point that we decide to call 0V an reference all voltages to that node. 

The relative voltages are the same V1 still has 15V across it in both simulations, R2 always has 5.625V (at DC deferentially) and R1 always has 16.875V deferentially across it, no matter where you place the ground

It does not matter where you place ground if there is no currents flowing into it (like if you have two grounds in the circuit. If there is one ground, you can place it anywhere and the circuit will function the same. 

If you don't have a ground, the spice simulation will have an error. That is because the spice simulator makes equations from the circuit (like a human would) and then solves them. A human nor computer can solve these equations without a ground,because the solution for the system of equations will be indeterminate.