If someone bored right through the Earth’s crust to the mantle, would it cause a volcanic eruption or what would result?

Probably not. In most cases, your hole would collapse without reinforcement before you hit the mantle. Or you dig a wide enough hole (think about as wide as it would be deep) to prevent collapse. Next factor: pressure. To get an eruption you need to find a pocket of high pressure. Now comes the tricky …

If we fall into a sea of liquid, how hot does it need to be for us to die instantly and painlessly?

Instantly is a matter of opinion, so let’s focus on painless. Temperature is tricky as you will have some build up prior to the point of execution, also orientation and speed on emersion can become significate. So let’s layout the most likely scenarios. Destruction of the body faster than nerves can fire and be processed …