Using 6V to 12V Boost Converter for LED Headlight on Vintage Motorcycle Stator

I think I've been reading TOO MUCH.  I had a handle on what I thought I was going to need and use but now I'm getting confused again.

I have an older design motorcycle.  It has a 6v system.  Headlight runs off the the generator (6v coil on a stator) and the rest of the lights run off the 6v battery, (charged by another 6v coil on stator) and the ignition is powered by a third coil on the stator.

I need to replace the lights.  I figured I might as well go LED.

Quesion Part 1:
For the headlight, wouldn't a 6v to 12v "booster" work?  This would be Alternating Current coming from the stator, right?  So I woul need a 6v DC to 12v DC booster, or should I go 6v DC to 12v AC?  Will I need a resistor?

Question Part 2:
The rest of the headlights are powered by the 6v battery as stated.  I want to put a 12v battery in the bike.  I've been reading that the generator should handle charging the 12v battery with a change of the 6v rectifier to a 12v rectifier.
Correct?

THANK YOU!!!

In order to change the battery to 12V, you will either need to change the generator to a 12V generator or use a 6V to 12V boost converter. You will also need to change the starter (if any) and everything else in the electrical system to 12V.

There are 6V high brightness LED bulbs that might be suitable for a headlight:
https://www.aliexpress.com/popular/6v-motorcycle-headlight.html
http://www.ebay.com/itm/2x-6V-P15D-12-SMD-LED...Bulb-Lamp-Dual-Light-/221222644958?rmvSB=true
etc.

I would guess that if you don't have a compelling reason to go to 12V, it will not be worth the hassle.

Question Part 1: Yes, but if the power to bulb is currently AC then you will want a bridge rectifier and filter capacitor to provide the input power to the boost converter.

Question Part 2: No, rectifiers are rated by their maximum working voltage. If the alternator/generator is generating 6VAC, using a 12V (or 100V) rectifier will not change the output voltage (it will typically actually go down slightly as higher voltage rectifiers typically have a higher forward voltage drop).