Computer Architecture: Is Electrical Engineering background necessary to became CPU designer?

Not anymore.  Introductory Computer Engineering courses will typically focus on the CPU's architecture and probably assign the students to create their own in a hardware description language like VHDL.  A solid understanding of digital logic is very important. That said, it is practical to learn the basics of electrical engineering if you ever want to work on embedded systems or other applications that aren't related to CPU architecture itself.

I don't think it is an absolute requirement. It helps to have peripheral knowledge as there are various aspects to processor design. Very broadly, there is an architecture aspect, a circuit aspect, and a process aspect to designing a CPU. If we go for the intrinsic, even mechanical engineering expertise is  required for designing the heatsink for the CPU. The process engineers at the fabrication centers for manufacturing the CPU in real silicon need know-how in semiconductor physics, fabrication technology, and possibly material science. It all depends on how deep you want to go down the rabbit hole. Computer or Microprocessor architecture is where a computer science graduate should feel right at home. Because this involves a lot of simulation of benchmarks, coding up any innovation and testing it against benchmarks to see how your cool idea impacted performance, and power consumption and bunch of other metrics.  A lot of ideas from OS, networks, compiler design, and distributed systems can be applied here, with a singular purpose of delivering the maximum performance at the lowest cost. However, if you are the curious type, you're going to want to know everything about how a CPU works.  And that is a good direction to go, because having knowledge of supporting technology will also help you to be a better computer architect.

As an opinion, I wouldn't say necessary, but it could be helpful. When it comes to basic Newtonian Physics, we have a standard set of equations. As you move up the scale to Light Speed, these equations are stretched out of all proportion, and as you move down to the Quantum level you're again dealing with distortions in the opposite direction, or in some cases completely new equation sets. If one places "Electricial Engineering" in a domestic household setting (as opposed to "Industrial Electrical Engineering" for big heavy equipment) the details to which one pays the most attention will change. In the Industrial Environment, where electrical loads are going to be heavy and long term, more attention will be paid to the size of the electrical conductor. In the Household setting, many manufacturers will cheat. Here in Australia, the standard electric kettle is rated 240VAC 10A (2400 Watts), yet many such appliances manage to get away with 7.5A cable instead of 10A. The incentive is to save money. They get away with it because a kettle is a short-use appliance and the cable can cope with 10A for a few minutes. On the other hand, should that cable become damaged, I'll go for over-engineering and spend the extra to replace it with 15A cable because I don't like the idea of power cords getting hot! This sort of thinking isn't going to work inside the construction environment of a CPU where one wants maximum conductivity through minimum material and ever-decreasing insulation gaps. When you look at an NE555 timer in its DIL package of 8 pins, it contains approximately 25 transistors and it's not uncommon, in these days of super-miniaturization, for people to use discreet components to reproduce the NE555 for education and experiment. When you compare that to a CPU with 500+ pins and several thousand transistors crammed into less space, it's not something that is going to be reproduced on a breadboard smaller than a city block. Remember that the modern "exposed" chip is around the size of a finger nail and must be in direct contact with a heatsink or all of the magic smoke will escape all in less than a second. This requires thinking at the opposite end of the spectrum, which is why I suggest it would be helpful but not entirely necessary.

Not just electrical engineering, you need to also have a deep knowledge of quantum physics and material science to design a CPU on your own. Not every electrical engineer will ever get the knowledge and chance to design a CPU since the technology is very advanced and requires huge investments and teams of scientists from many fields to work in parallel. And of course computer science is a must but not enough to get over all design challenges.