I think you use FP usually when in broad day sunlight, and you want to use a big aperture to isolate the object in focus from other, and after lowering the iso to the minimum and having the sync speed to the max, your picture is still over expose.
Say in very hot sunny day. You need shallow DoF. So you use Aperture f/2.8. Your shutter speed is 1/1000s with the lowest ASA. And you want to use your SB800 for fill in flash. Use SB800 in FP mode allow you to sync in 1/1000s.
Alternatively, you can use ND filter and hope your shutter speed will down to 1/250s or below so you no need to use flash FP mode. But remember, use any filter will degrade the picture quality .
1. Yes, you can still use FP mode while doing fill flash. FP mode is just a special mode on the flash to allow it to sync above the x-sync speed of the camera (1/250s for your D200). You can still dial down the flash eV to just fill-in the shadows or create a catchlight in the eyes.
2. Only necessary if the shutter speed needs to be above 1/250s on your D200, which sometimes can be the case if you're shooting outdoors with a large aperture.
Basically the maximum sync speed for any camera is the shutter speed at which the shutter is still able to reach full open position before the second shutter leaf starts to close. If you took a normal flash picture at above the sync speed, then the part of the frame will be blocked out by the already moving second shutter leaf (try taking a picture in a totally darken room and you will see what I mean, the faster you go, the larger the black portion you get under the frame)
However with high speed sync flash, the burst of flash is designed to stay on for the duration of its sync speed eg. 1/2000 sec (a standard flash lasts only about 1/30,000 sec).
Hope this clears your confusion.
No. The standard flash duration is variable form about 1/800s to 1/30.000s (number is not correct as each model is different). Flash in M mode for M/1 will be around 1/800s (depend on flash model). Flash in M mode for M/128 is around 1/30.000s (again, depends on flash model). Flash duration in TTL or AA or A mode depend on feedback received by flash.