Can hard disks read and write simultaneously on different tracks? How?

Is it possible that one read-write head can do both copy and paste at the same time even on different tracks? How does that happen?

I/O requests are put in a queue, and handled one by one.

There are various disk scheduling algorithms, picking the best one improves the time required to handle all the requests that are in the queue. A disk scheduling algorithm in general orders this queue in an attempt to more efficiently handle the request such that the head has to move less, the FIFO aka FCFS algorithm is an exception to this as it just reads the queue...

Given that the seek time (the time to go to a location) is smaller if the head has to move less fast, multiple requests are handled fairly fast using the right disk scheduling algorithm. This is why it looks like if it was read/written very fast, but not in any case simultaneous... :)

I'll list the most popular disk scheduling algorithms, more can be found at Wikipedia.

Note: Files are usually fragmented, so to read one file multiple requests are put in the queue.

Note 2: If you mean "why is moving a large file from one directory to another on the same volume so much faster than moving it to a different partition?" then the answer is because only the file's entry in the file table is altered rather than physically moving the file. — Comment by Amazed

FIFO aka FCFS: First-In First-Out aka First Come First Served

All requests are taken from the start of the queue without reordering them.

SSTF: Shortest Seek Time First.

The request that takes the least move of the head is taken from the queue.

Note how the head has to move less, it stays still for the first three images.

SCAN: Like an elevator.

This always take the furthest away, then handling stuff it passes along the way.

In this picture it starts at track 2 and plans to go to track 4; so, it tries everything at track 2, then track 3 (nothing), then track 4, after which it goes to the furthest request which is track 1; so, it goes back and tries track 4, track 3 (nothing), track 2 (nothing) and then track 1.

— Images: University of Wisconsin-Madison - CS 537 - Disk Scheduling (Originally might be by a book)

One step at a time.

1. Set k to 0.
2. Read from x+k.
3. Write to y+k.
4. Increment k by 1.
5. If not done, go to 2.