So for number 1 you are saying this is not good ?
No it is good technically. It is just that buying a fake Arduino does deprive the Arduino company of money that is used to do good things, like fund this forum and other educational projects.
Number 2 if I'll buy the led you suggest to me I will not need port extender correct ?
Yes you will not need port expanders for the outputs of your system. However, there are 21 pins on the Arduino and to do exactly what you said, use 20 sensors you will need one port expander or to find a way of reading more than one sensor per pin. There are ways of doing this using resistors and an analogue input.
Number 5 I will use the led you suggested to me and I am going to use 20 pcs 4-Pin WS2812B WS2812 LED Chip & Heatsink Board DC5V 5050 RGB WS2811 IC Built-in RGBW RGBWW WWA LED Chip
Also did you know that this sort of LED can be got in a ready built strip with three different numbers of LED per meter? Normally these are 30, 60 and 120 LEDs per meter so for 20 LEDs you would have the choice of three different lengths. Just search for 'ws2812b led strip' you can normally buy them by the meter.
I will have to calculate the voltage output for those and make sure the hall switched is good with them
No your sensor goes into the Arduino, it has nothing to do with the LED's voltage.
Number 4 are those good ? A3144E Hall sensor OH44E magnetic sensor switch Hall-Effect IC contactless
This is the tricky part of your project, the detection of the steel ball and the magnet. That switch will typically switch with a field of 55 Gauss but can switch as low as 20 Gauss. The big question is what sort of field will you get with a ball baring rolling along dragged by a magnet? This is one of those situations where you are going to have to try one, a lot depends on the mechanical construction. We have discussed other sensing methods here like foil contacts, I would like to throw another one into the mix, an optical detector do it switches when the ball covers it up. You might need a light above the detector to make sure there is enough light to turn it on when uncovered. I have used the SFH3410 in past projects for this. But I would start off by trying to make the foil contacts work unless you think the "magic" effect will be lessened.
Number 6 about the programming .......... (I need to do it by myself without help) and I am not going to cheat.
Glad to here that. However, asking for help is not cheating, asking for someone to do it for you is. I used to Lecture at a University in England and would be happy if my students were pointed in the right direction by someone but not if they got someone to do the work for them.