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Imagine we took earth and slowed it down so that a day lasted 100 years, could humans (or any life) survive? Let's assume that there are no ill effects of slowing it down, no massive tidal waves, no continents melting as they slide across the surface, and the atmosphere stays in its relative location. In short, every particle just decides to take a break from spinning. Would life be able to survive?

I have this image in my mind of people living on the band between the hot and cold hemispheres and gradually moving their settlements along with the rotation of the earth. In one regard, energy would seem to be relatively easy to harvest - just put a boiler in the sun-facing region and a condenser in the dark region and you could have a nice steam engine. Would people be able to use this easy energy to overcome the challenges presented by the scorched and frozen earth that surrounds them?

The big problem I see for survival is not energy, it is growing plants for food and feed. The dark side is going to turn into a desert, no grass or crops or trees or any photo-synthetic life is going to survive even a year of night, and there is no time for them to adapt. Which means all the wild animals and insects that depend upon plants are dead, and all the predators that eat those are dead.

You have a similar problem with the oceans; the photosynthesis that creates the basis of the food chain stops. Most of our oxygen comes from that activity, When the Pacific is on the dark side, you just created a low-oxygen planet, and that is going to kill trillions of fish.

Some plants will do fine with 24/7 daylight, and may adapt to it, but every year, 1% of these high-light plants move into night and will die there. Coming out of night into the day, there is no guarantee that what has become desert is going to suddenly sprout with life. Just as the Sahara was once green, but rain doesn't help it grow again.

Or, the constant sunlight may turn the bright side into a desert, too.

When the big agricultural regions in the US and Asia go dark, there goes the food supply for the world, and here come the food wars to figure out which 20% of the population gets to live on 20% of the former supply.

As for the energy, solar energy is not usable for all daylight hours, the sun is only high enough for about 10 hours a day. You will still have the dawn and sunset bands where insolation (that is the technical term) is strong enough to extract useful energy from sunlight. So similarly, only 37% to 42% of the slow earth, at any given time, has sufficient insolation to use for either photovoltaic power or solar concentration (thermal) power.

I don't think "wild" humans survive this. A high tech civilization could, the power generation could be mobile, crops grown indoors and given artificial night by simple shading mechanisms, oxygen generated the same way. But there is no huge advantage to having 24 hours of sunlight versus 9 or 10 hours, in terms of energy this is just a linear relationship. The bright side isn't going to boil, the atmosphere is a fluid and will act as a heat conductor, the flow will circulate hot air to the dark side, where it will cool, and that will inevitably push cool air to the bright side.

But the lack of photosynthesis: That's going to kill us all, we will be starving for both food and oxygen pretty quick, and there is going to be mass extinctions of wild life that may well destroy the ecosystem on Earth.

Over these long distances (dark side DS and bright side BS) trapping heat and conveying it from BS to DS is impractical because of heat losses along the way. You would rather harvest energy on BS and export it to DS.

Put it another way: a solar heater retains more heat than its surroundings. Or, solar mirrors will concentrate more heat than surroundings into a tiny area. This heat difference relative to the environment is sufficient to generate electricity and conduct it to DS.

Second way: this slow rotation creates an almost eyeball Earth where strong winds rise up and move from BS to DS, cool and descend and move back to BS. So, wind energy will be more available in DS and will not require moving any energy source from BS to DS.

In addition to what @ChristmasSnow said, 7.7 billion people "gradually moving their settlements along with the rotation of the earth", when 70% of the Earth is ocean, and a big chunk of that is the Pacific is -- at best -- impossible.

I'm dubious as to whether anyone could do it, since the ocean storms caused by those winds will be stupendous.