I have some of these flicker candles that use 3 AA batteries and I can use them but plugging them into an arduino, which happens to be there anyway, 5v and ground and they work fine. Now I have another that only takes 2 AA batteries and I am wondering if I can use some of these resistors that were in the newbie startup kit to reduce the voltage from 5v down to roughly 3v.
In this case I could, but while I will always have the power supply anyway, I may decide to use the Arduino for something else. That seems to happen a lot. I get tired of one gadget I made, stop using it and then use the device for something else.
I would also consider a resistor (1k or 10k) parallel to the LEDs to provide some load when the LEDs are off.
If you have a variable voltage source you can try measure the supply current while slowly increasing voltage. If the current does not increase it means a constant current source is driving the LED and it is probably safe to use the LED with 5 V. I would try it and take the risk but not everyone would do so because the LED may be damaged by this.
Not a good idea IMO, depending on the Arduino. Some Arduino boards (e.g. classic Nano) use the built-in 3V3 regulator of the CH340G chip. While this chip looks relatively big, in my experience it's quite flimsy electrically and overloading it can make it start corrupting data transfer.
It's a different story for boards that have an actual 3V3 regulator, especially if it's a somewhat more beefy AMS/LT1117 - and preferably not the once again quite flimsy 6202/6205.
Schematics of all Arduino's are published on their website.
Simple answer: when connected to USB, 5V passes through a Schottky diode to the rest of the board including the 5V pin(s). When connected to an external power source through the barrel jack, external power goes through an onboard linear regulator. Not on a Nano of course, as it doesn't have a barrel jack.
It's mostly intended for preventing damage to the USB port/computer in case a mild overvoltage condition occurs on the side of the Arduino. They call it the 'auto-selector' diode because it 'automatically' selects the power source the Arduino runs on - USB or 5V applied to the appropriate pin, without the external 5V source accidentally trying to power the USB port.
It's not a fool proof protection, but it's convenient as long as all voltages remain more or less within reasonable bounds.