While a world that follows the premise of the question has serious issues to face, just going along with it, I think you might find this wiki page helpful.
Hardnesses of the Elements (data page)
In such a world there would be alloys that we've never even thought of trying to make, and they would likely find some alloy that solved their problems. However, if we're sticking with pure elements, of particular note is rhodium, the first element on your list of precious metals, which has a Brinell hardness of 540, comparable to abrasion resistant steels used in the mining industry and much harder than things like cast iron or even tool steel. It is also durable, so it doesn't wear easily. It's a noble metal, and like all of that family is is very non-reactive.
It's not perfect, of course. Its biggest drawbacks are that it is brittle and that it has a high melting point.
I don't know if anyone has ever simultaneously possessed enough rhodium and had the desire to make a hammer or axe head out of it. In its pure form it can't be made into good jewelry because it's tool brittle and the jewelry breaks. That said, a large lump of it is a different question, and it would definitely be possible to chop down a tree with an axe with a rhodium head. Breaking rocks? The pick/hammer might shatter or chip, depending on the hardness of the rock. However, again, it should be comparable to AR steels, and they're used for mining purposes.
It melts at 1538°C (3565°F). This is above the temperature that a coal/air forge can reach, so it couldn't be cast using coal to melt it. To be clear, it would be able to be heated to workability in such, and since it is typically found as an uncombined metal (like gold, basically) it should be able to be forged into a lump and then further into whatever shape is needed. It is apparently quite difficult to machine - there are numerous machinists complaining about it ruining their tooling on forums.
Neither better nor worse really, but worth considering, is that it is also ~1.5 times denser than iron, so any tools would be heavier (or smaller).
Finally, if weapons are a consideration, it would be fairly awful for most of them. It would potentially make a ferocious projectile weapon (arrowhead or bullet, that is) but a large, chunky sword or axe would be too heavy for human scale muscles to use effectively and a thinner weapon would be too easy to shatter. I guess that might actually pose an interesting challenge to duelists - not quite as bad as fighting with glass swords, but a similar idea.