When people talk about plate tectonics and such, they're talking about events that take place on geologic timescales. Plates move about as fast as your fingernails grow. If we consider the kind of movement between that would've taken place between the height of Uruk (one of the great Sumerian cities) and now, we're talking about plates moving about 0.08 miles (416 feet, 5,000 inches) at the low estimate, double that at the high estimate. And neither of those figures is remotely meaningful on a timescale of 5,000 years.

So, as for your specific examples. Beringia (which is the name given to the possible "land bridge") emerged around 20,000 BC as the sea levels fell due to widespread glaciation. This date is really fuzzy and it probably happened earlier, as the last glacial maximum was another 5,000 years before that. As the glaciers retreated (ie, melted), they freed up much of the water that had been removed from the sea and the sea level rose. Glaciers started to retreat in North America no later than 16,000 BC and in Antarctica no later than 12,000 BC, with Beringia probably ceasing to be a viable land bridge within, at most, a couple thousand years of the latter date. After all, if your two major sources of glaciation are melting, that's going to seriously impact your sea levels.

As for the Morocco-Iberia land bridge, that is a significantly older event. The Mediterreanean Sea had effectively been cut off from the Atlantic c. 6 million years BC. After a few hundred thousand years, the land bridge that acted as a de facto dam of the Mediterranean was broken and the basin was flooded again. And while it's nice to hypothesize that this might have inspired all those Great Flood stories, our nearest human ancestor at the time was probably an off-shoot of ardipithecus, which predates the split between humans and chimpanzees. So... yeah, it's really not likely that any stories survive from that time.

So, had the people of Uruk had access to satellites and cutting-edge cartography, would the world they have seen look fairly similar to ours? Sure. There would be geographic differences. Today, Sri Lanka (which is nowhere near Mesopotamia but bear with me) is an island. Back in the time of Uruk, it would've been an island connected by a land bridge to the Indian subcontinent. That particular land bridge was destroyed a little over a decade before Columbus' famous voyage, so, while a relatively significant geographic event in the region, it's also relatively recent.

Then, of course, there are the ways that we humans have impacted the geography of our world. We dig canals, erect bridges, build irrigation systems, cut down trees, blow up or carve our way through mountains, and generally do all sorts of things to bend geography to our whim. And, in that respect, the landscape the people of Sumer would've seen probably looks rather different from what we see. Plus they didn't have to deal with modern urban sprawl.