Why not look on the datasheet? There are maximum guaranteed worst-case ratings and graphs showing typical spec's.
Yes, that's the very point that I'm making. I ask because would there be any overhead available for a 317 to regulate with? (i.e. after the battery drops to 1 V per cell)
Beware though, many an engineer has found reality on the breadboard different to data sheet interpretation believed as gospel at the time. I've had engineers try to tell me the datasheet is wrong, after they'd replaced a "faulty" IC! Until pointed out that datasheet data is often percieved out of context. The manufacturer often shows performance in the best posisble light, i.e. how low their 'drop out' is, it looks impressive until you read the small print down the bottom of page says tested at only 1 mA.
I have the feeling an LDO would be better than a 317.
LEDs do not work in parallel unless their voltages are matched. Their voltages are very different
Voltages matched to what? (I think you mean 'currents need to be matched' don't you?, the voltage being a function of colour, current and LED chip temperature) Dramatic proof, connect a red LED in parallel across a lit green LED, the green LED goes out completely.
In practice, when it's only low currents same-colour LEDs 'kinda work OK' connected in parallel. But when the LED's are run hot, as one warms up it takes a bigger and bigger share of current, because its diode drop voltage falls as LED chip warms up, I think that's what you were saying?
Anyways, LEDs for illumination tend to be better connected in series for this reason.