Electrical – Voltage and Amperage for LED Cube

If you put all these LEDs in series you'll indeed need 38V, plus a bit for the series resistor. But you can make a circuit with several branches. Put 2 LEDs in series and you'll need 7.6V. So the remaining 9V - 7.6V = 1.4V is the voltage over your resistor. If you want 20mA through your circuit You divide this 1.4V / 20mA = 70 ohm, so a standard 68 ohm will do nicely.
Now that's for 2 LEDs, you could go for more if you had a higher voltage. the way to calculate the resistor is the same. If you want 10 LEDs from 9V, in theory you could put the circuit 5 times in parallel. That would be a total current of 100mA though (5 x 20mA), and that's a bit much for a 9V battery.

edit
It's been suggested that you could replace the 5 resistors by a single one, and branch from beneath this. In an ideal world that would be true; the resistor value would then be 1.4V / 100mA = 14 ohm. But this isn't an ideal world, and there may be small differences in LED voltages. In that case the branch with the lowest voltage will draw most of the current (100mA!) while the other LEDs will hardly light at all.

You are correct - the forward voltage depends on the forward current.

The forward voltage you see in the table of typical values is for a current of 20mA, which is too high when all 3 colors are used at the same time (footnote two in the absolute ratings table on page 3 - 15mA is the maximum in that case).

When you look at diagram 2 in the data sheet, you can see the relation between forward voltage and forward current. What you see here is that for a forward voltage of 3.3V, a forward current of 20mA can be expected. With 3V, it would be 8mA. A higher resistor value doesn't make this more reliable, it just makes the LED darker. You want to have the resistor as small as possible.

The resistor should be only large enough to drop the forward voltage to about 3.1V with a current of 15mA - this would mean a value of about 13.3 Ohms (the one for the red LED needs to be larger, though).

Whether this LED is usable for you depends on the brightness you need. If you don't need it to light up fully (or you use a version with higher intensity, see page 4), it would work. If you want to be sure you can use the full intensity, you need to use another one. Olin is right - the variation between batches can also mean that some are brighter than others. To ensure a uniform brightness, you need to control the current flowing through the LEDs.