Thanks to the kind and speedy response of Amanda, I could overcome most of the troubles I had.
My two main mistakes in the beginning were the gauge and hooks size which came out too small with the yarn I have used. I gradually used a bigger hook and doubled the thread, till I finally arrived to 8mm and 10mm size hooks!
Another mistake is that in Row 3, there is apparently an error in the instructions, saying “in next ch2 space” (instead of “in the same ch2 space). This is why I didn’t get any increase for so many trials.
Finally, with the yarn I used, for a 2-4Y size (maybe it is even 24M), I did as follow:
8mm hook with 11 sts in rows 1-2.
10mm hook with 22 sts in rows 4-8, 44 sts in row 9, 12sts in row 10 up to row 13. Then sl-st all the rest of the hat, excluding the brim.