But as we burn that sugar, the Sun keeps supplying the same amount of heat as it was while the sugar was being created.

Imagine a river, a damb and a reservoir.

The river flows to the reservoir and through the damb. All of the water came from the river, but if we release too much from the reservoir at once, the damb will overflow.

Correct, however.

Energy in = energy out + energy stored

When the Sun is creating sugar the Earth can reject the remaining heat. But when we start converting that sugar into heat, the Sun does not turn off, it keeps supplying the same amount of heat as before. So now if we take the constant supply of heat from the Sun and we add the conversion of stored sugar energy into heat, without increasing the Earths capacity to reject heat, at a certain rate of sugar burning we will overwhelm the Earth's heat rejection capacity and the internal temperature of the system will rise.

Humans as all warm blooded creatures generate heat and the planet has a limited capacity to reject that heat - thermodynamics dictate that the Earth has a limited carrying capacity for humans. This is putting aside all the pollution that you mentioned that we create, which will likely kill us way before we cook the planet with our farts and sweat.

Gravity doesn't need to be fought. Spin up a cylinder of 500m in diameter at 2 rpm and you have earth gravity. Luckily, in space that cylinder will keep spinning.

For radiation shielding - 5m layer of dirt (formed from asteroid mining slag) solves that issue.

Saying we will never live in space is like saying we will never: circumnavigate the globe, fly or go into space.

Sadly, living in space is easier to imagine than everyone throwing their plastic bottles in the trash and not on the ground.

Side note - by modern standards planetary colonization and terraforming are silly sci-fi ideas born from a time of the space race. It's akin to contemporaries of Jules Verne describing the future where everyone takes a personal biplane to work at their floating blimp office.

The more likely scenario based on modern or near future tech (which may be as silly as above examples by the time we actually get to that point) would be building artificial habitats in space. A megastructure 3/4 the size of our moon could house roughly a trillion humans. These habitats are imagined to have an artificially maintained 24 hour day/night cycle.