Are there any species that would be able to sustain themselves on the sunlit face of the planet, the habitable Terminator-zone or even the freezing icy side of the planet? How would animals adapt to these environments? If needed, what genetic tweaks could be made to make some animals perfect for surviving in this new alien ecosystem?
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A tidally locked planet does not need to have the extremes of the scorching hot sun-facing side and nitrogen freezing cold side.
So you could have a planet which still has fairly "normal" living conditions. A planet that isn't super close to a star and has a thick atmosphere would be able to "regulate" it's temperature quite well.
But assuming you have a tidally locked planet with extreme conditions (such as Mercury) and animals similar to how they are on earth I would assume to following:
Surface temperatures of a couple of hundred degrees mean that animals would need to stay underground to not die. The only way an animal could live on the surface is if it would have some kind of highly reflective "umbrella" big enough to not only shield itself but also the surrounding ground.
Depending on how big your terminator is (the bigger the better), normal life as it is on earth would be possible.
Surface temperatures constantly under the freezing point of water and even air means that animals would need to stay underground so that they do not freeze.
I would suggest you watch this video from Isaac Arthur about humans colonizing Mercury. The video goes into great detail on what human life on such a planet would look like. These circumstances are the same for animals so you can adept human solutions to animal ones.
I would argue that any animal that can borrow underground will do great on this world. Flight wold be very difficult due to strong winds generated by the difference in temperature so bird like creatures are very unlikely to form. Cold blooded animals would exist but would be dominated by their warm blooded counterparts due to the fact that they cannot leave the warm side of the life belt. With the exception of flying animals I think you can have most of the animals that you have here on earth (but even those can exist).