Could a planet with a core of majority tungsten and cobalt with traces of gold,copper,iron and uranium be able to support life?
There is no reason why such a planet should not support life.
The planet would need to be at the right distance from its host star, and the right size with the right atmosphere. There would need to be a decent mixture of elements in the crust. The core has little to do with the habitability.
Any conductive, convective, rotating core can generate a magnetic field. There is no need for iron to be present (the iron is well above its curie temperature, so no permanent magnet can form). With a magnetic field, the atmosphere should be safe from the solar wind.
However, there is a reason why most of the Earth's core is iron, and not tungsten. Iron is much more common in the universe than tungsten. There is no way that a tungsten core could form (barring magic)
Firstly, a planet that forms via the accretion in a stellar nebula disk is going to form our of the material is present in the disk. To get a core super rich in tungsten there obviously needs to be a bunch of tungsten in the disk. This is an issue because all elements heavier than Iron are formed in Type II Super Nova explosions. Fusion of elements heavier than iron produces no net energy. So when larger stars start fusing Iron, after going through all the elements up to it, they collapse because there is no longer sufficient radiation pressure to hold up the star. This collapse forces fusion to create all the elements heavier than Iron.
This process is messy, that means If you got enough tungsten out of such a supernova to make a planetary core, then there is a whole bunch of other elements ahead of tungsten that were made. And after too for that matter. This means that during its accretion the planet would likely not have picked up enough carbon, oxygen, hydrogen and phosphorus to support life because it formed in a very metal rich part of the cloud. The lighter elements would have been blasted further than the heavier ones, adding more difficulty to accreting them.
Forming in such a matter means it probably will miss out on getting an atmosphere that even remotely resembles earth (or early earth). So natively, it will probably be unable to support life. Terraformable, yes but supporting life outright no.
Also, there is the supernova remnant to consider with is either a neutron star or black hole. If this planet is still gravitationally bound to the object, it may be subjected to dangerous levels of high energy particles.
We think the required feature of the core is convection. That is the movement of the molten (metal) material due to heating/cooling. Probably, what is necessary for that is a inner solid (hotter) core and a surrounding electrically conductive liquid (metal).
Our outer core's temperature isn't well understood. While the outer outer core is probably around 2700-4200°C, the inner outer core is somewhere between 3700° and 7700°C. Tungsten and cobalt alloys probably melt between the mp's of the two elements, and cobalt melts at ~1,500°C; W at 3,500°C at 1 atmosphere; at the much greater pressures inside the Earth, I don't know how much those numbers will increase, but you could figure a 1:1 alloy would be 2,500°C mp, and increase that by 50% at high pressure. I don't think the experimental information is available, it's certainly not readily available.
So, in my opinion, you can use this "back of the envelope" calculation without being called out. It's a little "hand-wavy", but still. Just say that the composition contains W+Co+trace elements which are liquid and since you've gone that far out on a limb, just include the fact that the solid core is ferromagnetic (due to its composition and enormous pressure).
The above answer is provided by assuming you meant "develop/evolve" life, rather than "support" life. Life exists on the scum and muck that is a pretty good description of the surface of our planet. What's needed for life to exist (i.e. be "supported") is
- the correct nutrients
- trace nutrients, and lots of C, N, O, S and P and Si as well as the trace elements (nutrients) we need
- the temperatures and pressures have to be "mild" and there can't be "too much" of any elemental poisons.
The subsurface is only important if there is significant seismic events (volcanoes, and other venting - from, say, plate tectonics) which allow
- surface materials to be subducted
- materials deep in the mantle to be brought to the surface.
It may be that plate tectonics must accompany core convection, but that's another debatable question. I don't see how it matters either way. The real issue is the assumption that something like tungsten, W, which has a natural occurrence of 0.17 ppm compared to iron's 300,000 ppm would be found in enough quantity to build a planet out of it.
If you've got the ability to build the planet up to the regolith, then the light dusting of ecologically necessary materials on top is trivial. There's no clear reason why a core of lead, or of silicon, or diamond, or tungsten, or osmium couldn't "support" life.
I am not a geologist, but I think that more important are materials in surface layers. And also, it depends on your idea of life form that should inhabitant such planet.
If you will insist on humanoid (and generally Earth-like) life forms, then my answer will be NO, but else there I don't see why such planet (with core from materials you wrote) should not be able to keep life.