Since this answer hinges on Gus' interpretation of events, we must look at it through Gus' eyes. This is a long explanation, be warned!
What we know about Gus
Gus is a civilized man. He behaves with self restraint, is intelligent, does not gamble or overplay his hand, only takes the reliable path. He dislikes violence, and only chooses to use violence tactically, not out of anger or a sense of retribution.
Although he is diplomatic, he dislikes those who are violent, like Hector Salamanca. He also dislikes those who operate without being intelligent enough to do so (to Gus' standard), which is the core of why Gus dislikes Jesse.
Gus meets Walt and Jesse
Walt is an intelligent and civilized man. He is not a street criminal. His genius is further proven by the quality of his product.
Jesse is a white trash junkie. He doesn't think before he speaks, is considerably dumber than Walt or Gus, his addiction makes him a liability.
However, seeing these two people makes it very clear why they work together. Walt is the genius cook, and Jesse has the street connections for their business to take off.
Seeing an intelligent and civilized man running a successful business must be a rare sight in the world of meth. It's not clear whether Gus saw Walt as a kindred criminal spirit, but it stands to reason that Gus must like Walt as a business partner, compared to the far more common and more uncivilized street criminals like Hector Salamanca.
Similarly, Gus chooses Gale (a newbie cook) over a presumably long line of experienced but uncivilized meth cooks. Because Gale conforms to Gus' standard of civilized crime.
However, Walt's methods are not as clean as Gus'. Comparatively, Gus' operation is much larger, but also considerably better due to the increased reliability and decreased risk. He thinks of himself as a better future for Walt's genius.
Gus really dislikes Jesse
Gus dislikes Jesse to a fault. He assumes that any bad behavior on Walt's part is a consequence of having to deal with the likes of Jesse. Gus does see that Walt needed someone with connections, so he understands the necessity of Walt associating with Jesse.
But he wants to elevate Walt. Not so much as a matter of doing something good, but because Walt could make him a lot of money of course.
So he offers Walt the following:
Step up to the big leagues, I will give you everything you want. But leave Jesse behind, he is only holding you back.
To be fair, it is a really enticing offer, from a business perspective. But Walt declines.
Walt even goes so far as to suggest that he pays Jesse's share out of his own pocket. Not just pay him, but split the money 50/50!? To Gus' mind, that is ridiculous, Jesse does not do half of the heavy lifting of Walt's enterprise. A white trash junkie does not deserver 50% of the payout.
Now, we have to make a distinction between the truth and what Gus assumes.
In truth, we know that Walt keeps Jesse around because Walt does not consider Gus as a friend. Jesse is loyal to Walt, not Gus, and his involvement gives Walt more executive control over the meth cooking (compared to when one of Gus' men would have helped Walt, e.g. Gale).
It's also possible that Walt considers the fact that Jesse has access to Walt's personal life. And if Jesse feels unfairly cut out of the deal, then he might take petty revenge on Walt. Although Gus would always crush Jesse if he acted out; that does not mean that Walt wants to risk Jesse spilling the beans to Skyler in any way.
But that is not what Gus interprets from Walt's declining.
Gus still recognizes Walt's genius. And he also recognizes that Walt is fairly new at being a criminal. So Gus interpret Walt's desire to include Jesse as a form of misplaced loyalty, a sign of Walt's gratitude towards Jesse for helping him become a meth cook.
Gus is still trying to establish a relationship with Walt, so he begrudgingly acquiesces. He allows Walt to involve Jesse; only because it furthers Gus' goal of having a fruitful business relationship with Walt.
Jesse acts out of line about Tomas
Jesse raises the topic of Tomas, and actively creates conflict between Gus' employees: the dealers on one side, and Walt and Jesse on the other. Jesse forgoes the chain of command and acts on his own; which goes against Gus' principle of a neatly run business. The chain of command is there for a reason.
Jesse is confirming what Gus already thinks about him. He is led by his emotions, not his intelligence. He acts before thinking. He is a liability who could do serious harm to the business without even understanding it.
But Walt protects him. Other than trying to make Walt let go of Jesse, he cannot do anything about it right now.
Furthermore, while Walt does side with Jesse, Walt talks to Gus about it. Walt is also much more open to a compromise, whereas Jesse is not. This appeases Gus. It shows respect for the business, and a pragmatic approach. Walt therefore also confirms what Gus already thinks about him. He is clever, pragmatic, respectful, open to compromise; not at all like the street criminals that Gus despises.
Gus sees Walt as someone like him. Gus very much prides himself on being a manager: his respectful diplomatic behavior, and his desire to find suitable compromises.
Gus therefore agrees to find a compromise. Again, the only benefit to Gus is that he gains favor with Walt. Although I do think that Gus must have personally disliked using children to deal drugs. Not only is it morally reprehensible, but children are a liability (they will often get away with no punishment if they confess their crimes, making them liabilities).
But this time, Gus gives a direct order. No retaliation against the dealers, as it upsets Gus' business. In return, the dealers no longer employ children.
Tomas is killed
Although this must have irked Gus too; I do think he is pragmatic enough to see that it is the cost of doing business. "No loose ends" is exactly his business style.
I don't think he ordered to have Tomas killed, but he understands the dealers' decision. It was an impossible choice to save Tomas, it would risk the entire multi million dollar enterprise.
At any rate, Gus is only informed of this after the fact; so he can't undo Tomas' death. The best course of action is to let bygones be bygones and not upset his business. Keep in mind that the dealers are technically speaking Gus' customers. He can't keep upsetting them and still expect them to work with him.
Jesse retaliates against the dealers. And Walt helps him
This is where Gus' opinion of Walt goes off the rails.
Although Gus can understand Walt's loyalty to Jesse; Walt has also disobeyed a direct order from Gus himself. This irrevocably means that Walt puts Jesse before Gus, to a point of adding risk to Gus' enterprise.
This cannot be tolerated. Walt has proven to not respect the chain of command when it suits him. He is a liability, as much as Jesse is.
Whether Walt has already been a liability (and Gus didn't see it), or Walt has become a liability because he learned from Jesse; doesn't matter at this point. Just like Walt picks Jesse over Gus, Gus will pick his business over Walt.
Gus cannot stand to kick Walt out immediately. Walt is also not a current threat; it's more a matter of Gus now being aware that Walt will act out when push comes to shove.
Therefore, Gus sets the plan in motion to hire Gale and slowly phase Walt out of the meth cooking; so that Walt becomes expendable and can be done away with.
I don't think that Gus wanted to kill Walt, but I do think that Walt's personality had made it very clear to Gus that Gus could not remove Walt from cooking without Walt retaliating. The same principle applies again: Gus does not want to uspet his business; and therefore considered the necessary evil of killing Walt.
As to the evidence you provide in the question:
1.Walter only killed those 2 workers of Gus to save Jesse, and there wasn't any way of correlating those kills with Gus's business, so it didn't raise suspicions to Police Investigation;
True, but it is the principle of the thing. Walt broke the chain of command, and even went so far as to ignore a direct order. Walt also refuses to acknowledge that Jesse should be punished for this. This is unforgivable for someone who runs a tight ship like Gus.
If another similar situation presented itself, Walt would do the same thing again, and Gus does not want to risk that.
- Walter could be useful to Gus as an informer about Hank's investigation.
True, but the benefit of what Walt can pick up from Hank does not outweigh the detriment (risk) that Walt's actions introduce. It's not worth it.
- Walter never betrayed gus: he dodged all the times the investigation of Hank related to the drug business.
That is not the only way in which Walt could betray Gus. Walt betrayed Gus by breaking the chain of command, ignoring a direct order; and siding with Jesse over Gus to a point where it affects Gus' business. This is not acceptable to Gus.
Walter only cared about him and his family, but I think he was very discreet on what concerns the drug business.. Even so, Gus wanted to kill him. Why?
Walt does not only care about him and his family. It is a quintessential part of the plot that, while Walt argues that it's only about securing financial stability for his family in his absence, he actually is making a statement that the world has shat on him unjustly, and that he is a force to be reckoned with, as opposed to the sad little high school teacher that he had let himself become.
