This is the link of what I have in mind, https://www.youtube.com/watch?v=jKvnn1r-9Iw , can a planet rotate chaotically through this system and still be habitable? If it's going to collapse after a period of time, how much time does it take for the planet to fall into something or drift away?
First, the figure-of-8 orbit, as pictured, is a solution to the three body problem. However it is not a "stable" solution, in that if you disturb the orbit by a tiny amount, the orbit won't remain in a figure of 8, but rapidly become chaotic, and end with either one sun colliding with another or being ejected. This is an example of the "butterfly effect". Since it is impossible to avoid disturbing the orbit, there will never be a figure-of-8 orbit in the universe.
Next, if a planet is moving "chaotically" then pretty soon it will either be ejected, or collide with one of the suns. This would happen pretty quickly. (the actual number can't be calculated, but it would be a matter of years, not millennia)
The only three body systems that are stable are
- One large star, with two smaller stars orbiting it (like a star with two planets.)
- Two distant stars, with a third star orbiting very close to one of them stars (like a star-planet-moon system)
- Exceptional co-orbital configurations such as stars at Lagrange L4 or L5 points.
Any other orbital configuration is not dynamically stable.
If you go ahead and have a system like this, you will have to deal with extreme variation in the amount of sunlight the planet receives, and consequently, there will be massive and unpredictable variation in climate. This is a hard planet for life.
The whole point of the three body problem is that there is no general solution that proves the stability of a certain orbit. The number of terms it takes for the power series analytical solution to converge is beyond our capacity to compute, at least for now. Given that we cannot prove the orbital stability of the planet in your setup, it is hard to tell what the conditions on that planet would be.
As @Aron pointed out in comments, there exist numerical solutions to the n-body problem (as well as many families of general solutions). But a numerical simulation (using a program like Rebound, though presumably more advanced) alone only proves the stability of a single scenario with all the inputs provided to the simulation. In 'real life,' there are nearly infinite small gravitational perturbations, caused by motion in other planets, uneven density distributions in the main star, passing comets, etc. Without modeling these inputs exactly as they happen over billions of years, the numerical solution cannot prove stability. Thus, without a general analytical solution, we cannot prove stability of a real-world n-body orbital setup.
As far as we know, it takes billions of years to develop intelligent life. While the probability of a planetary life-cleansing event (like, say, plunging into a star) is very small on a per-year basis, I conjecture that on the timescale of billions of years, the likelihood of a life-cleansing event happening to a planet in a chaotic orbit are high.
So the answer is: we don't know the answer to your question, but I, at least, think the answer is no.
Yes it can, but the inhabitants will have to come from somewhere else. They will have to bring everything with them to set up their environment, right down to the microbes. Finally, they will need to stay ready to leave quickly in case their unpredictable planet starts heading towards something lethal.
On the positive note, the inhabitants will probably be able to hide from the rest of the universe, since no one with enough science to understand evolution would expect to find intelligent life under such conditions.