We received a solution to the first part of the problem from Andrei of School NO. 205 in Bucharest and from Chong Ching, Chen Wei and Teo Seow from Secondary 1B, River Valley High School in Singapore. Well done to you all.

The diagram below shows how the students from River Valley High School combined the original tetromino, together with 15 copies of itself, to cover the eight by eight chessboard.

Andrei followed this up with: It is clear that the 4 in a line tetrominoes can be fitted into an 8 by 8 square, because they can be fitted into a 4 by 4 square:

Here there are a lot of possible arrangements, all leading to the same result.

The L-shaped tetromino can also be fitted into an 8 by 8 square for the same reason:

Here there are also a lot of solutions.

The square tetromino will obviously fit into the 8 by 8 square, but there is only one possible arrangement.

Finally, I checked the possible combinations for the Z-shaped tetromino, but, as I analysed the possible combinations, I observed that there weren't any.
Now, I must prove it.
I tried several combinations, then I observed that there couldn't have been any solutions because there always remains a place where it isn't possible to put a tetromino. For example:

or

This is the only tetromino that will not cover the 8 by 8 chessboard.