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.
Sorry, but no. Attempting to drive an LED with a constant voltage is a great way to kill an LED. The coefficient of forward voltage with temperature is negative, which means that, driven by a constant voltage, if the LED starts to get warm it will start to draw more current, which will increase the power dissipated by the LED, which will cause the forward voltage to drop even more, etc to disaster.
You are better off assuming an LED forward voltage of about 2 volts, then provide a convenient voltage/resistor combination to provide ~ 1 mA, then measure Vf. Then reduce the resistance so as to produce a series of currents in the 1-10 mA range, and at each current setting check the encoder for proper operation.
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Have you tested the two alkaline batteries? I'd bet they are slightly above 1.5V each, bringing your total voltage up to the LED forward voltage. However, when the LED starts conducting, its low impedance pulls the battery voltage down slightly, and the battery itself acts as the current limiting resistor.
As to the second part of your question, I can't say whether its always better to add the resistor and use more cells. If you're making an LED throwie, then it definitely isn't. If you want a consistant runtime that doesn't rely on hitting a bullseye between battery voltage and current draw, then probably.