What is ground and what does it do?
The first time I tried to do the current calculation for a circuit similar to the previous circuit on a simulator, the program complained about not having a ground and "floating voltage sources".


Your simulator wants to be able to do its calculations and report out the voltages of each node relative to some reference, rather than have to report the difference between every possible pair of nodes. It needs you to tell it which node is the reference node.  

Other than that, for a well-designed circuit, the "ground" has no significance in the simulation. If you design a circuit where there is no dc path between two nodes, though, the circuit will be unsolvable. Typical SPICE-like simulators resolve this by connecting extra resistors, typically 1 GOhm, between every node and ground, so it is conceivable that the choice of ground node could artificially affect the results of a simulation of a very high-impedance circuit.


  I picked ground at the bottom, but would it be okay to pick ground between the 7 ohm and 2 ohm resistor - or any other place? And what would be the difference when analyzing the circuit?


You can pick any node as your reference ground. Often we think ahead and pick a node that will eliminate terms for the equations (by setting them equal to 0), or simplify the schematic (by allowing us to indicate connections through a ground symbol instead of by a bunch of lines connecting together).


  I've read that there are 3 typical ground symbols with different meanings - chassis ground, earth ground, and signal ground. A lot of circuits I've seen used in exercises use earth ground or signal ground. What purpose is there in using earth ground? What is the signal ground connected to?


Earth ground is used to indicate a connection to something that is physically connected to the ground beneath our feet. A wire leading through the building down to a copper rod driven into the ground, in a typical case. This ground is used for safety purposes. We assume that someone who handles our equipment will be connected to something like earth ground by their feet. So earth ground is the safest circuit node for them to touch, because it won't drive currents through their body.

Chassis ground is just the potential of the case or enclosure of your circuit. For safety purposes it's often best for this to be connected to earth ground. But calling it "chassis" instead of "earth" means you haven't assumed that it is connected.

Signal ground is often distinguished from earth ground (and partially isolated from it) to minimize the possibility that currents flowing through the earth ground wires will disturb measurements of the important signals.


  Another question: since the ground is at unknown potential, wouldn't there be current flowing to or from ground to the circuit?


Remember, a complete circuit is required for current to flow. You would need connections to earth ground in two places for current to flow in and out of your circuit from earth ground. Realistically, you'd also need some kind of voltage source (a battery, or an antenna, or something) in one of those connection paths to have any sustained flow back and forth between your circuit and the earth.


  However, when there are multiple voltage sources, some of them are "floating". What meaning does the voltage of a floating voltage source have?


If I have voltage source with value V between nodes a and b, it means that the voltage difference between a and b will be V volts. A perfect voltage source will generate whatever current is required to make this happen. If one of the nodes happens to be ground, that gives you immediately the value at the other node in your reference system. If neither of those nodes happens to be "ground" then you will need some other connections to establish the value of the voltages at a and b relative to ground.