Is the water I drink the same water I pee? Or does it get taken apart and put back together with different pieces?
Most of the water from metabolism is released in your breath. Water you pee is mostly water you ingested via drinking and eating.
The water that leaves via your lungs and the water that leaves via your kidneys all had to get there via your blood. There's no distinction made based on the "source" of that water.
And the reason for that is that air is bad for heat exchange and transport, water is much better, and evaporates! While the body passively cools in air (either directly when air is colder or via sweat evaporation when air is too hot) most of the heat exchange happens actively via breathing, which is nice because it's easy and quick to adjust, just by breathing faster or slower, so as to maintain 37ish C-100ish F while the body is continuously burning molecules. Without breathing we'd not only lack comburant (oxygen), we'd overheat pretty quickly!
As you mentioned, sweat is also evaporating water, and that is a much more effective way of dumping heat.
It is possible to recycle urine into drinkable water. That’s what they do in international space station where water is very precious.
https://www.nasa.gov/mission_pages/station/research/news/wat...
But it's not just sewage though, it's chemicals as well, e.g. from medication, plus whatever else gets flushed.
In Singapore's case the sea is the obvious sink for treated sewage, which kicks it back to desalination. Except that I believe that Singapore's water comes across the causeway to the mainland in a pipe.