I'll try to contribute, trying to omit what others have already mentioned;
The first issue I want to mention is your idea that an asteroid would 'go through' Earth just as the bullet does through the high-tech jelly. This analogy is deeply flawed due to one simple thing that you have ignored:
There is actually a lot of physics involved in stopping something when hitting something else, but it can all be broken down to a rule-thumb:
For high-velocity impacts, if the density contrast between impactor and impactee is high, one will ignore the other. If not, both break apart. (Over at astronomy we think a lot about impacts)
What do I mean by this? See the bullet is much denser than the medium it is shot into. But 'much' I'm talking about around a factor of 5-10 density contrast. After the initial expansion shock visible in the animation, the gel medium more or less ignores the bullet.
The same thing would happen If you shoot a bullet into water (well, because it has nearly the same density as the gel),or air (duh, but the principle still applies).
Now if you shoot it at something even denser, like a brick, or a metal plate, the bullet will not penetrate anymore.
Now imagine the bullet is a meteor and doesn't have the same material strength as the bullet, then it will simply fly apart. Large rocks ($\ge 100 km$) in space have comparable densities, so they will always be ripped apart, never penetrate.
Smaller rocks will always be ripped apart, without affecting Earth significantly, because they have similar density, but the small rock also has much less momentum.
I mentioned this to maybe guide your thinking in physical terms: Density contrast and momentum (but that was already mentioned in other answers).
I find most of your questions that you had in the comments in your physics.se answerable just by looking at this principle.
If you want to, this can be a valuable tool in world-building, as well as in bible studies, depending what your aim is.
So what you want to do is essentially send a mini-black hole through earths crust and still keep the solar system and the planetary atmosphere in a good shape after that. That would be a tough one.
For your other questions, I will just briefly comment on, as correct answers already exist:
1.) An asteroid hit at an angle to slow the Earth's rotation and to increase its axial tilt to give birth to seasons?
2.) An asteroid passed through the crust like a bullet causing little damage externally but changing the magma flow to decelerate the Earth's rotation? It would still leave a creator but the bulk of the impact would be absorbed by the inside of the Earth.
3.) An asteroid struck such that the polar caps melted or the water was forced out of the oceans?
1.) As others said, that's not possible. But I think the other answers underappreciated how NOT POSSIBLE this is.
To go from a 1.7 hrs rotation period with the whole mass of Earth to 24 hrs in one impact would deposit a lot of kinetic energy in the crust. So much in fact that the energy deposited would easily reach the gravitational binding energy of the planet.
Meaning: This would be an impact that could rip the planet apart. It wouldn't just be catastrophic, apocalyptic or hyper-bad. It would be fatal. I can show you the math if you're interested, it's simple.
And as soon as you go down in impact energy, you can't produce the slow-down anymore that your source is citing.
I have discussed 2.) above already.
3.) is an entirely possible scenario, and probably happened multiple times during Earth's early history, during the Late heavy bombardment period.
However it is important, that this happens before the development of complex life, or else everything has to start from zero again. To give you a sense of comparison, the K-Pg impactor (the Dinosaur Killer, around $\rm 10 km$ in size) wouldn't have near enough energy to affect the polar caps significantly or to evaporate any ocean.
To do this, you need a bigger meteorite (or asteroid with $\rm \ge 100km$ size, as you asked for) and then consequences are again dire:
The splash energy (and as discussed, the density principle needs it to be a splash) would be incredible, melt the whole surface, even eject a good part of the atmosphere mechanically.
But only after that comes my favourite part: Atmospheric escape. The extremely hot surface would heat the atmosphere to temperatures where it can escape in bulk.
No impactor scenario that I can think of fulfills all the possible consequences that you need in your story.
- Of course you can leave out the part with changing the rotation period of the planet, because that places the hardest physical bound on a planet-destroying impact.
- The 'going-through-Earth' thing would also have to be abandoned on a realistic ground. Except you want to invoke really many rogue neutron stars / black holes flying around in the universe and hitting Earth by chance.
- Melting the polar caps / evaporating the oceans should be fine, but then you have to live with the other consequences as well. Or you magic them away.