Can I use a flash on a Sony a7c that, during bright daylight I would
be able to make the background black/low key (like a black wall - the
ambient exposure ) while the subject to be correctly exposed ?
Assuming full sun falling on the subject and the wall? With a really big flash (like a 600-1200 Ws studio strobe) it might be possible, but it would be a lot easier with smaller lights in the shade or in a studio setting rather than outdoors in direct sunlight. To kill the ambient, you need to light at about +5EV above the ambient.
How important is the flash sync speed and max shutter speed?
It's important if you need faster shutter speeds while shooting flash, either to keep from overexposing in the ambient, or to achieve a thin DoF with daylight fill flash.
There is a flash feature called high-speed sync (HSS, aka focal plane or FP flash) where the flash can "fake" being a continuous light source for the duration of the exposure by pulsing continuously. But this feature also lowers the power output of a flash by about -2EV or more.
There's also a another form of syncing that I call "tail syncing" (commercially it goes by hypersync, hisync, HS, supersync, etc.), where a flash is typically fired at full power (longest pulse) with a slightly delayed shutter so that the flatter more even tail end of the pulse can be used as a continuous light source for a faster (1/1000s or so) exposure. So there are ways to use flash with a faster shutter speed. They're just going to rob you of flash light output as well and can be a game of diminishing returns.
If all you need is thin DoF with daylight fill, you can also consider putting a neutral density filter on your lens to allow you to shoot at sync speed or slower shutter speed without overexposing.
Should I skip a7c and go for a7iii instead?
I honestly don't think it makes that much of a difference for this specific feature. All focal plane shutter camera bodys have a flash sync speed limit. And while 1/250s is faster than 1/160s, it's only by 2/3 of a stop. And you'll still have to use HSS/FP flash or some other technique to get around that limit.