You are correct the voltage in a car will fluctuate, so the value of the resister needs to be a little bit over 1000 Ohms to allow for this fluctuation [say 1100 (or 1.1 K ) ohms.
The LED may change brightness a little as the car voltage fluctuates, but the eye doesn't notice changes in intensity much less than 50%, so you may not notice the fluctuation in brightness or if you dod it will be small.

For LEDs in series, each LED will "consume" (cause a voltage drop of) about 1.7 volts, so for 3 LEDs in series, the voltage drop over the LEDs will be 1.7*3 = 5.1 volts.
This Voltage drop across the LED is called the "forward voltage" and is often quoted as a useful charateristic of the LED as some LEDS have a much higher forward voltage [like 3.4 volts].

The forward voltage (plus a little bit) is also the lowest voltage that the LED will work (light) with, so a normal LED won't light below about 1.7 volts, so thats why they don't light up when connected to a single battery cell, which is about 1.5 volts or so.

So you now with 3 LEDS in series have to rework your calculations for the original resistor and since the voltage the first LED will "see" is now 5 volts, you take the difference between the car voltage and the "voltage drop" (i.e. 7 volts) and plug it into the equation again to get the revised resistor values.

If you use the existing 1000 Ohm resistor for a 1 LED circuit and then put 3 LEDs in series, then either no LEDs will light, or they all will, but dimmer than with 1 LED in the circuit.

Either way, you won't blow anything up, so suck it and see.

Don't ask me what happens if you put them in parallel - probably they all suck the same amperage, so the resistor needs to be made smaller to allow for the higher current.