First, to reach the ocean, is it more practical to melt down through the ice, or drill it?
The Russians drilled a hole 4 km down to Lake Vostok. They used a drill. The technology they are using is prefectly capable of drilling a similar borehole on Europa.
Would it be practical to build habitats mounted on the underside of the ice, in the water, or would drilling rooms within the ice be easier?
In this question I calculated the water pressure at the base of a 20 km thick layer of ice at 237 atm. Since hydrostatic pressure scales linearly with depth, at 10 km thick, pressure would still be 118 atm, equivalent to 1250 m in our ocean. Modern submarine are rated to a pressure of about 500 m. Assuming that transportation cost of materials is a significant factor (i.e. moving all that structural alloy from another moon/asteroid, then down a 10–20 km shaft), it is probably not worth making a large permanant living structure that deep. Humans need a lot of air space at 1 atm to live comfortably, and that is very expensive to make available.
Also in that other post, I calculated radiation exposure. 10 m of ice is really all you need, so there isn't a lot of value in going too deep.
If melting/drilling a tunnel down to the ocean, would it be possible to passively leave this borehole open, or would the pressures cause the ice to flow together again?
Ice has a viscocity. From this textbook, the viscocity is about $2\times10^{13} \text{Pa}\cdot\text{s}$. This means that an internal pressure of $2\times10^{13} \text{Pa}$ will impart an expansion of 1m/s to an ice mass. 118 atm is about $1.2\times10^{7} \text{Pa}$, so the imparted speed of expansion will be $\frac{1.2\times10^{7}}{2\times10^{13}}= 6\times10^{-7}\text{m/s}$ which is about 5 cm per day.
The pressure of the ice on any structure made to hold the tunnel would be about 12 MPa. That pressure isn't excessive, but since ice is viscous you can't just put some support struts in there. The ice will ooze around it at 5cm a day. To put a cylinder in to maintain the size of the hole you need it to be.. well at least 10 km long. Too expensive.
The Russians drilling in Lake Vostok have similar problems here. Their hole is only 4 km, but since gravity on earth is higher, pressure is higher too, up to 350 atm at the bottom of the ice. They don't use a structure to maintain the hole, they simply melt all the ice that seeps in with a mixture of kerosene, freon, and antifreeze, and then pump it out.
This solution is a bit harder on Europa. Lake Vostok itself is about −3C, while the surface temps can be as low as −89C (coldest place on earth, incidentally). However, they don't drill in the winter so −20 to −50C is more like what the drill team sees at the surface. The surface temperature of Europa's surface is −160C, but the liquid ocean would be warmer than Vostok, based on the phase diagram of water and anticipated pressure of 12 MPa.
What kind of engineering would be required to keep the borehole open, assuming my colonists want an elevator to the surface.
After drilling is complete, the hole is kept open by pumping an anti-freeze solution around the edges of it to melt it. The anti-freeze interacts with the ice, lowering its lowering its freezing point below whatever temperature the ice is at. The rate of ice encroachment is relatively small, but since the shaft is big, the amount of ice to be removed is large. Assuming 2.5cm on average of ice encroach along the entire 10km length of the shaft, and with a 4m radius borehole, 12600 m$^3$ of ice must be removed every day, or 12 million tons of it. Fortunately, by melting the ice with anti-freeze and letting it flow to the bottom of the hole by force of gravity, the flow rate of 0.14 $\frac{\text{m}^3}{\text{s}}$ is not unrealistic. That would take 4 standard, 3" firehoses, and is about 50% more than you can get out of a single fire hydrant.
The biggest engineering challenge is removing the ice that is collapsing in the top half of the ice sheet, where the temperatures are closer to −160C than zero. No anti-freeze is going to work at those temperatures; car anti-freeze freezes at -40C, and alcohol at -110C. The anti-freeze itself will freeze. Some sort of heating system will be needed. It would be much more effective, once the borehole is dug, to maintain it from the bottom, since that is the warmer side of the ice, and since gravity will pull melted ice down into the warmer regions without need for pumping; you only have to pump antifreeze and not melted ice too.
So you basically have to pump your heated anti-freeze solution from the base of the ice sheet to the top, and recover it at the bottom, separating the antifreeze out for reuse, and presumably dumping the water/ice into the ocean.
Pumping up is a big issue, due to shutoff head limits for centrifugal pumps. I deleted the math as extraneous since this post is already forever long, but, suffice to say, a centrifugal pump, which is good at high volume pumping, will not get the pressure you need. However, any good pressure washer can get the pressure you need (3000 psi = 204 atm) and they do this with positive displacement pumps. So you will need some enormous positive displacement pumps; flow rate has to be relatively high or your heated antifreeze will cool and freeze before it reaches the service. Not an impossible engineering challenge, since I have seen them. If you want 200 gpm of 3000 psi reciprocating positive displacement pumps, you will need about 400 kW of electrical power, based on the pumps I've seen.
So that brings us to generating both a.) enough heat to unfreeze a 10km hole and b.) enough power to run a 400 kW electrical load forever; for reference this is what a 100 kW diesel generator looks like and c.) doesn't take a ton of fuel. The solution with today's technology is a nuclear reactor. Fortunately, they already have them in submarines, so it's not too much to ask for to bulk up the pressure hull to handle higher pressure, and install one at the bottom of the ice to keep the hole open. Though, keep in mind, you can't assemble it 10 km under the ice, so the hole has to be big enough to get the thing down there in the first place. Also, you have to replace it every 10–15 years once it runs out of fuel.
In conclusion
Most people would permanently live in habitats dug a few meters into the ice. This would give plenty of room for expansion by digging more warrens into the ice, without having to go into high pressure areas, and also keeping the colonists close to the outside world.
The hole would have to be significant. A submarine style pressure hull with reactor would have to be inserted into the hole. However, since the smallest nuclear submarine had a pressure hull about 4 m in diameter, the size doesn't have to be unreasonably large. Maybe an 7 m hole and a 6 m pressure hull with nuclear reactor and ice melting equipment. This could be operated remotely, its not a threat to human life if the hole closes if there are no people below the hole. The worst you have to do is re-drill the hole.
In fact, I don't anticipate humans going down the hole at all, too dangerous. Just some construction-bots to install your ice-melter and some submarine-bots to explore. Maybe an Alvin for exploration, but you'd never want to try to dock and transfer people to the ice-melting hull at 12 Mpa.