I like your idea here, but I don't think it's possible.
First issue...If Mars does in fact have a molten core, a large part of its molten nature is derived through pressure and not heat. The article you linked produced a molten core at 40 giga-pascals with a minimum sulphur content of 10.6%. Lacking the pressure or the sulphur content, this core is solid. If it was possible to drill to it and have it rise up the hole you drilled, the pressure on it would reduce and the material would be back to a solid before it ever had the chance to flow on the surface (mind you, I'm not too familiar with the physics behind drilling to the core of Mars). Remember the paste is semi-liquid due to pressure more-so that heat.
Second issue is the gasses actually being released by this. Sulphur dioxide (formed in high heat magma) would readily solidify when exposed to Martian temperatures...however Hydrogen Sulfide that's formed in low temperature magma (like this would be) would remain a gas except around the polar regions. Hydrogen Sulfide is exceedingly toxic...most safety detection equipment considers 5-10 ppm of Hydrogen sulfide enough to set off an alarm...50-100 will cause serious eye damage...300-500ppm can disable our nervous system...and 1000 part per million of Hydrogen sulfide is near instant death (collapse of lungs after a single breathe). It's actually been used in war times.
Of the gas you'd see escaping into the atmosphere from an eruption...very little of it is what I'd consider human friendly, with the exception of water...you may have to begin a refining program to remove everything you just finished releasing. That said, a good amount of CO2 would get released from this and you may have the opening for plant life to have an abundance of what it requires.
Honestly, I think you would be better off attempting to thaw the polar regions and look for various gasses hidden away under the frozen ice caps, not unlike what is under Earth's permafrost.
Hard to speculate on if there is enough water to form an ocean...remember that mars is relatively flat (no plate tectonics to produce mountains or hills) and the majority of the land is the same elevation, with the exception of volcanoes. Olympus Mons being the largest there. An ocean on mars covers most surface area leaving just these volcanoes sticking up (admittedly Olympus Mons would make a pretty large landmass). I would think that much of Mars's water has been lost over millions of years, though I can't completely rule out a presence of underground water.