The reasons why there are so many plot holes in DBZ anime is because it was filled with fillers that were not in the manga. The fillers became nonsense and to help replace scenes that they took out to make sure that the episodes fill the time slot. The manga is much better with the storyline. Most of the dragonball z movies are not canon meaning that they do not fit in the dbz timeline. It was just a bonus.
I know the movies are not canon but it seems that Frieza is the younger of the brother and it seems that he was like a brat. The little kid who thinks he is the toughest and must have everything and if he doesnt get it he will throw a fit. Its apparent in the scene where he came back as frieza bot. his conversation with his father kind of portrayed him as a spoiled kid.
Even if Cooler exist as canon in the dbz timeline, it would make sense that he was the older brother and willing to spoil and give his little brother what he wants.
The reason why Frieza have such a fearsome reputation is because he's the only member of his family blowing up planets and destroy galaxy. People who sense his power will know that he is strong and fear him. It didnt seem that his father ever did any of those things even if he is more powerful than Frieza so nobody knew about it.