Are the traces of a PCB always made of copper, we can use another conducting material? Also, is there is another conducting material that can be used in the cladding layer instead of copper, or do we always use copper clad?
The PCB process generally assumes copper conductors.
This is somewhat by definition because all the equipment is designed around a copper etching and plating process.
And the technology - the common set of equipment, standards, practices, and knowledge - also generally assumes copper process.
This is the "technology brand" known as printed circuit boards aka PCBs, for now at least.
The reasons for this are described in the comments, copper conductors are superior for many applications.
That being said, other conductors can be printed/plated/etched and otherwise attached to various substrates.
For example, here is a silver plated process with ENIG finish on ceramic substrate produced by Remtec
They certainly look like PCBs to me, but the manufacturer does not refer to them as such.
There are also "aluminum trace PCBs" (to differentiate from "aluminum base PCB", which use aluminum as a heat-sinking backplane) but they are more expensive than copper PCBs.
It serves a very niche market: superconducting traces for cryogenic electronics, transparent PCBs for X-ray systems, PCBs with long radiation length materials for nuclear beam applications.
Pro: very good for aluminum wire bonding.
Con: it is difficult (although possible) to solder to aluminum using fluxes based on fluoroborate. However, it is also possible to have copper plated into the aluminum to make it easily solderable.
For extreme low-cost low-power applications as pocket calculators carbon traces printed on plastic sheet are a thing. They also function as keyboard contacts.
The high resistance of carbon isn't a problem as the currents needed for keyboard matrices and LCDs are in the 1 µA range.
I have seen flexible printed circuit heaters that used nichrome instead of copper. However, these are pretty specialized and are designed for a single purpose: to act as electrical resistance heating elements. They are common, but specialized in the sense that they are only really good for one purpose: heating.