Auto mode is auto everything, exposure and flash and ISO and white balance and color profile, everything. Cannot turn auto features off, so no choice of selecting anything. Auto is auto.
In M,A,S,P modes, the user controls these things, as desired. Some individual auto features can be chosen, but they can also be turned off, to select specific values to use.
There is no Flash Off option in M,A,S,P modes. Flash is auto in Auto mode, and Auto mode needs the Flash Off option.
But the way M,A,S,P works is that if you don't want flash in M,A,S,P, then you simply do not turn the flash on, or simply do not open the internal flash door. Then flash will be off. This is the photographers choice to do, and how it works.
The camera manual instructions for flash with M,A,S,P modes starts with "turn the flash on, or open the internal door".
The instructions for Auto mode says "don't open flash", the flash will be automatically used when deemed needed (but Flash Off will disallow it in places where flash is prohibited or objectionable).
There is a 1/60 second Minimum Shutter Speed With Flash in A and P modes. Indoors (dimmer light), if you want a shutter speed slower than 1/60 second, you have two options. Use camera M or S mode, and set any slower shutter speed directly. Or in A or P modes, you can use flash sync options of Slow sync or Rear Curtain sync, and that will use the slower shutter speed actually metered.
In M and S modes, shutter speed is what you set it to be.
In A and P modes, shutter speed is metered, but the 1/60 second Minimum applies, unless Slow sync.
In Auto mode, the flash and shutter speed will remain completely auto, and Flash Off is the only choice to disallow it. Flash Off is only needed by Auto mode.
For slower shutter speeds:
Example, with flash turned off, maybe the dim scene meters 1/4 second.
(speaking of Nikon).
You turn on the flash, and in A or P modes, the shutter jumps to 1/60 second Minimum Shutter With Flash, because 1/4 is too slow to handhold. This is NOT a value metered from ambient, ambient will be underexposed. But you are using flash instead, you normally don't need 1/4 second.
In A or P modes, if you set Slow Sync (or Rear Curtain sync, which includes Slow Sync), then the original metered 1/4 second will be used (whatever ambient meters will be used).
Or you can use camera M or S mode and set any slow shutter speed you desire.
Shutter speed does not affect the flash exposure, but it will seriously affect the ambient exposure. Overall photo exposure is the sum of ambient plus flash, so we gain added control when we realize that flash photos are a "double exposure" combining the two exposures. Indoors, ambient level is often insignificant, but is the overwhelming factor for fill flash in bright sun.
More expensive Nikon models have an E2 menu where this Minimum Shutter Speed With Flash can be chosen (any speed not faster than 1/60 second).
If you want a shutter speed faster than 1/60 second, then either go into a brighter light scene that will meter higher in A or P modes, or use camera M or S mode to set what you want.
But in M,A,S,P modes, you enable the flash on when you want flash to fire, and don't when you don't. If the flash is "on", it will fire. It is not enabled automatically except in Auto mode.
1/200 second is that camera's Maximum Sync Speed for flash. That value is typical and most common. It cannot be faster to properly sync flash. Some camera's and speedlights offer a HSS flash mode (converts instantaneous flash to be on continuously for awhile, thus no sync requirement) to allow any faster shutter speed. Mainly, the purpose is that this also allows a wide open lens with flash outdoors. However, HSS has serious disadvantages of allowing only around 20% flash power, and HSS also bypasses the speedlights motion stopping capabilities.