If the earth's core is made up of copper/other metals other than iron-nickel alloy, how our magnetosphere will be?
On a quick approach:


Magnetism. The copper itself have a weak magnetism, so a copper core will not create a magnetosphere. Chek here or here.
Gravity. The Iron density is 7.874 g/cm³ and the nickel density is 8.908 g/cm³. Copper density is 8.96 g/cm³. So with those density data, the core will be heavier. (The actual core is supposed to have a 9.9-12 g/cm³ density) It is supposed that this additional mass will increase the gravity. What it is not well determine is the % of iron and nickel at the core, so, we can talk on probably a raising on between 5-15% the actual gravity force.


Regarding some other metals, it will depends on magnetism and density.

Hope it helps!