A mushroom is a heterotroph, meaning it obtains its nutrition by consuming organic matter from other sources. It cannot produce its own food through photosynthesis like autotrophs.
That would just be fungi in general, but it could also be Heterotrough.
Yes, producers do break down dead organisms through a process called decomposition. Decomposers such as bacteria and fungi play a crucial role in breaking down dead organisms and releasing nutrients back into the ecosystem for producers to utilize. This recycles the organic matter and allows for the continued growth of living organisms in the ecosystem.