If our solar system is travelling 230 km/s how is it possible for Halley's Comet travelling 70.56 km/s to leave our solar system and then turn up again 75 years later if it's travelling more than 3 times slower?

It's a matter of relative speeds. The 70.56 km/s is the speed of the comet relative to the sun. The 230 km/s speed of the solar system is relative to the center of the galaxy.

A similar question: How is it possible if the speed of the earth at the equator is 460 m/s that a ball thrown vertically at 20 m/s can come back down to where it was released?

The simple answer is the speed at the equator isn't 460 m/s.

Irrespective of what these speeds are measured relative to shouldn't we see tails trailing every object we see? Just like Halley's Comet they are all in motion.

The earth is part of the solar system. The earth is moving at approximately 460 m/s +/- the speed it is traveling about the sun.

Halley's Comet is moving at approximately 460 m/s +/- the speed it is traveling about the sun.

I am moving at approximately 460 m/s +/- the speed the earth is traveling about the sun +/- the speed due to rotation of the earth +/- the speed I am wandering around on the earth.

You are in an airplane that is called the solar system and the comet went to the bathroom and will be back in the seat soon lol. But all passengers and the bathroom and everything are still flying away without anyone inside having a clue without looking out.

Halley's comet is not traveling at that speed by the way right now (maybe at the closest approach to the sun it was that fast briefly). It is very slow now and for decades, slower than the distant planets or like 2-3-5 km/sec type thing. Earth is around 30 km/sec. But the solar system around the galaxy and the galaxy expanding are common velocities to all participants in our solar system. Its like you are in car and move your hands around and grab some apple to eat all while traveling at 110km/h. You do not leave your hands or the apple behind in the highway. They are still inside the car.

The tail is caused by ice evaporating when the comet gets too close to the sun, it's noting to really do with its speed.

It doesn't leave.

Then what is the speed?

That's not how this works.

https://en.wikipedia.org/wiki/Comet_tail