For this answer I'm assuming that Planet Plane is the one from this answer which is infinite in area and 6,378km deep.
(This cannot be Planet Minecraft because it would not have a horizon for the Sun and Moon to drop below. For what Planet Minecraft would really be like, see this episode of It's Okay To Be Smart.)
Firstly; would this hole immediately collapse or would it be a somewhat stable structure?
This depends on the material Planet Plane is made of. If we assume Planet Plane can exist, then your hole will not collapse.
The problem with massive structures is gravity tries to pull them together into their most stable state: a sphere. Since Planet Plane exists as an infinite plane then we can surmise it is strong enough to resist gravity. Therefore your hole will remain open for a while.
Eventually it will close. Any debris which falls into the hole will collect at the center where gravity balances out. Eventually enough debris will fall into the hole to fill it up.
The problem isn't will the hole stay open, the problem is will the hole grow forever?
As @Samuel pointed out in the comments, unlike a sphere where the center of mass is a point in the middle, the center of mass for a uniformly dense infinite plane with finite depth is an infinite plane in the middle of the planet. This means gravity is always pulling orthogonal to the surface
Side-view cutaway of Planet Plane
<------------- center of mass ---------------> P
If you punch a hole in this structure, there will be no gravitational force trying to pull the sides together. Every side will have the same amount of mass around it in every direction. However, the hole has no mass (discounting the negligible effects from air) and exerts no gravity. Therefore each side will be a little closer to one half of the plane than the other and so will feel a stronger gravitational pull in that direction. The edges of the hole will experience a tug from their side of the planet, threatening to widen the hole. The hole will experience tidal forces.
If the hole gets wider, the sides get further apart, and the tidal effect gets stronger. Once it starts, the hole continues to get bigger forever. Fortunately there's infinite planet, so you'll never run out of surface to run to.
I'd be interested to see someone do the math on the tidal forces at the edges of your 1km diameter cylinder.
Secondly; (assuming it was/made stable) if I had a death wish and took a running jump off the edge, where would I fall? The exact center of the hole? Would I drift across to the other edge? One of the "sides" of the hole?
The issue here is whether or not Planet Plane is rotating. If it's not, and if we assume the hole contains a vacuum, as you fall you will continue to drift across the hole at the speed you were running when you jumped. If you were an Olympic sprinter you'd be going about 10m/s and smack into the other side of the hole in about a minute and a half hitting it with the same force as if you had run into it, plus getting dragged along the edge of the hole. Ouch.
If Planet Plane is rotating there will be a Coriolis force and you will smack into a wall on your way down. Minute Physics has a great video on this.
The one exception is it your planet happened to be rotating on an axis running directly through your hole. This is the same as drilling a hole through the Earth pole to pole.
The best case (for you) is if the planet is not rotating and your hole have an atmosphere. In this case air resistance will slow your forward motion and you will eventually fall straight down. But this causes a new problem.
That same air resistance will slow your fall. When you pass the center point and begin traveling "up" towards the other side of the planet you're now feeling the same pull of gravity down into the hole. With no air resistance this will exactly balance out your acceleration due to falling and you will reach the peak of your fall right at the lip of the hole on the other side and you can gently step out.
But air resistance will be slowing you during your fall and ascent. This means you'll come up short. Instead of peaking at the surface, you'll peak below it and fall back again to the other side, again peaking further below the surface. You'll oscillate back and forth like this, losing energy to atmospheric drag, until you're stuck floating in the center of the planet.
Tidal forces also play a role. If you're in the exact center of the hole you're fine, but if you're closer to one side or the other the tidal force will slowly pull you towards the edge. As you get further away from the edges the force will get stronger. Again, I haven't done the math on how strong.
Corollary: If the hole itself was stable, would travel through it unprotected be fatal? Eg: are the pressures on a mine-craftian world the same as on a spherical one?
This depends on how old Planet Plane is, how it formed, and what it's made of.
The interiors of spherical planets are initially very hot from the residual heating of forming due to gravitational collapse, plus the decay of any radioactive elements trapped in the insulated interior. There can also be heating due to tidal effects as happens on the moons of Jupiter and Saturn, the squeezing and stretching adds heat to the moon's interior.
But Planet Plane can't be a moon, and it didn't form from gravitational collapse. So it's anyone's guess as to what heat of the interior will be.
What will be a problem is air pressure. Air pressure is the weight of a column of air pressing down on you. Assuming a normal Earth atmosphere at the surface, at the center of the hole you'll be supporting an extra 3,189km column of air. This is a lot of air. The atmosphere ends and space begins at about 100km. At the center you'll be experiencing at least 30 times normal air pressure. This is roughly equivalent to being 300m under water (every 10m is about 1 atmosphere of pressure). Squish You will need some serious deep sea equipment to survive this, only five people have gone that deep on the Earth.
(NOTE: I'm playing a bit fast and loose with the air pressure calculation. The pull of gravity will decrease towards the center of the hole which may reduce the air pressure. Still, it's doing to be in the ballpark of 30bar.)