5-10 mins.
I am following this video: DIY Electromagnetic Flashlight - YouTube
My magnet is only a half inch diameter compared to the 3/4 used in the video.
I guess I need more coils.
You will need a capacitor the size of a mid-sized sedan.
The homie Grant Thompson used a slightly larger magnet and generated 4V A.C, used a 1F capacitor and was able to get 10 mins of light from 30s of shaking
Great, so do that.
You will need a rectifier to charge the capacitor. Most people use a supercap. I usually recommend to stay far away from Instructables, but this one is OK:
Actually, AFAIK all the commercial versions of this use a Ni-MH battery!
I gave up on them as pretty useless. 
Yes, pretty useless.
20 years ago my company credit card company offered premiums or credits or cash refunds. For Christmas gifts, I got a case of those flashlights for gifts to friends and relatives.
When I inspected the flashlights, they would light with NO shaking. Shaking did make them brighter. further inspection and disassembly found 2 coin cells, non-rechargable, in series to light the LED. I guess they might take a little recharge, but not much.
I notified the CC company of the fraud and got my money back. Still have some lights in original boxes.
So the whole idea is an exercise in futility.
You can always increase the led current limiting resistor until it only lights when it should.
I am currently using a bridge rectifier to convert the A.C to D.C current but was told they have a significant voltage drop. I am using one that looks like this:
Would it make any difference for the voltage drop if I were to use one of these smaller versions?
I read something about how the schottky diode has a lower voltage drop. Is it possible that the smaller bridge rectifiers use this diode making it more applicable to my project? I am working with very low voltage and current and need to save as much as possible.
The smallest bridge rectifier would make the most sense for a flashlight (make your own from 4x1n4148 diodes), but the voltage drop will be about about 0.6 to 0.7 V per diode or 1.2 to 1.4 V for the bridge.
Schottky diodes have much lower forward voltage drop, but they have much higher reverse leakage, so the charge on the capacitor won't hold for long.
Add more turns to the coil.
Would there be any advantage of making the rectifier from 4 diodes rather just using the unit which houses the diodes anyway? Are there any advantages with the voltage drop in the smaller bridge rectifiers compared to the larger ones?
Probably not, in either case.
Using a bridge to convert to DC will reduce efficiency.
Oh dear.
Faraday’s law states that a current will be induced in a conductor which is exposed to a changing magnetic field.
Not a voltage - a current.
the LED is not frying because the current is low.
You DONT need one of these as youre unlikely to get 200 amps.
to a bridge made from 4 of these would be better
BAT85 30V 200mA 0.2A Small Signal Schottky Diode (Pack of 25)