I was reading about how Mercury's core makes up most of the volume of the planet. I'm guessing this is both because of the small size and the distance to the Sun. If Mercury got too hot could it completely "liquidify"? Would it join the Sun and completely merge with it, gradually crack apart and spread, or something else entirely? Would it cause other problems in terms of life on Earth?
I was reading about how mercurys core makes up most of the volume of the planet. Im guessing this is both because of the small size and the distance to the sun.
There are at least two hypotheses about the causes of Mercury's composition:
Early in the Solar System's history, Mercury may have been struck by a planetesimal of approximately 1/6 that mass and several thousand kilometers across. The impact would have stripped away much of the original crust and mantle, leaving the core behind as a relatively major component.
Alternatively, Mercury may have formed from the solar nebula before the Sun's energy output had stabilized. It would initially have had twice its present mass, but as the protosun contracted, temperatures near Mercury could have been between 2,500 and 3,500 K and possibly even as high as 10,000 K. Much of Mercury's surface rock could have been vaporized at such temperatures, forming an atmosphere of "rock vapor" that could have been carried away by the solar wind.
But if the second hypotheses is right Mercury should have been relocated to wider orbit later. Planetary migration is very interesting topic and I would say it's not solved completely yet.
At current orbit Mercury is not hot enough to evaporate silicates at any significant rate.
If mercury got too hot could it completely "liquidify"?
A lot of "hot jupiter" exoplanets have benn observed in recent years and results are very interesting. For example HD 209458b "Oziris" has a comet-like tail of evaporating atnosphere. Clouds of iron and titanium dioxide were observed too. Iron rain (literally) could be one of exciting effects.
Compared with "hot jupiters" Mercury is much smaller with less gravity. If closer to Sun it undoubtedly would have comet-like tail of evaporating elements, mostly silicates and metals. If intensive enough the tail could be seen by eye with black glass during Mercury transits (also the transits would be much more often at closer orbit).
At closer orbit Mercury should be tidally locked. The Sun-faced surface would be completely melted lava, but dark side would probably be solid because of lack of stable atmosphere. Evaporated rocks would cool down by thermal radiation and dowfall behind the terminator line.
Would it join the sun and completely merge with it, gradually crack apart and spread, or something else entirely?
The planet should be affected by some force to migrate even close and close to Sun. I can't tell surely will evaporation slow down the planet and move it closer to Sun but I suppose it will not. I suppose Mercury will not fall to Sun but will be evaporating. I can't do math now to estimate how close to Sun it should be to evaporate completely in less than 5 billion years (after that Sun will become a red giant and will envelope Mercury by its atmosphere in "coup de grace" act :) ).
I don't see reasons for Mercury to crack apart if it is not close to Roche limit (again with assumption that evaporation only is not enough to lower its orbit).
Would it cause other problems in terms of life on earth?
I don't think so. Maybe Solar system space would be more dusty because of Mercury's evaporating tail. Even cracking of Mercury would not affect Earth because the debris would stay close to Mercury's orbit (much closer than current Mercury's orbit, don't forget!). Theoretically some of debris pieces could develop more eccentric orbits in millions of years to reach Earth (because of collisions, uneven evaporation, orbital resonanses) but I bet all them would evaporate before that.
P.S. Most of this answer consists of educated (or not so educated) assuptions. So objections, amendments, corrections are welcome.