Almost "any" flash will work on an outside sunny condition. You can fire your tiny built-in flash and use as a fill light. That would count as portraiture.
Portraiture normally means you can put your lights near your subject, so, again a small flash with a "normal" softbox (40-80 cm) could provide you with an f/8 - f/11 easily, which is about the same range of a cloudy day.
Also remember than the diference between outdoor and indoors is given mainly because the ambient light which is controlled with the speed. So, in my opinion the high speed sync is very handly.
the power requirement for a monolight vary based on where and when I use it?
This is a little obvious question. But it is not about outdoor-indoor, it is about distances. Are you taking a wedding photography but you need to light a distant building as a background?
Is 600WS enough to shoot outdoors?... Also, given that the monolight is so powerful
You answered yourself.
Is there any downside is using it indoors or in any other place where the full 600WS isn't needed?
Is overexposed or a very small aperture enough downsides? This specific flash has 1/256 minimum power, so that is good.
I have the feeling that you have not experimented a lot with a combination of flashes + ambient light.
Buy a "cheap" speedlight + radio trigger combo, and play with it. Bounce it on a wall, on a celling, at day, at night.
But the short answer to the question regarding the flash is yes, that flash will work on outdoor and indoor.
"Is there a preferred or recommended wattage for outdoor and indoor strobes?"
No. It does not depend ond indoor-outdoor, depends on distances+area of coverage. You could need a verey powerfull sets of flashes to shoot a car or truck in an indoor studio because the area of coverage or a powerfull flash outdoors to use it far from the model. Or a small flash if you shoot near the subject in both cases.