Korra and Mako only started a romantic relationship after everything had been solved. By then, they weren't participating in pro bending matches any more. Mako never dated any one on the team during the pro bending tournament so he is taking his own advice.
At that time, there was absolutely no evidence that Hiroshi was a member of the Equalists other than Korra over hearing Hiroshi say something on the phone that sounded suspicious. It was a bold accusation to make and it didn't really make any sense. For that reason, Mako thought that Korra was jealous of him and Asami because it seemed like the only possible reason for Korra to make such an accusation. Yeah, Mako was being rude to Korra but he was trying to defend Asami and her father.
Mako wasn't being overly obsessive. He is protective by nature, and by that time it seemed like he really loved her. If one of my family members was kidnapped, I would act exactly the same as Mako. Asami didn't really seem to show much concern, I mean she was there but instead of wondering where Korra might be, she worried about why Mako was showing so much concern. To tell the truth, Asami and Korra aren't that great of friends so she didn't feel as worried.
At the end of episode 9, Bolin and Asami were standing at the back so Mako didn't push them away. He just told Lin to give her some space. Can't really blame him for doing that, I mean how do you expect Korra to answer a detailed question if she is half concious and exhausted.
Do you know how worried Mako was when Bolin was captured by Amon? How in any way did Mako not show concern for his brother? If he wasn't concerned then he certainly wouldn't have infiltrated an Equalist rally or gone searching for his brother. How did Mako not protect Asami and Bolin? Was there a certain situation where Mako left Asami for Korra during one of the fights? Did Mako leave Bolin for Korra during a dangerous situation?
Now as for Asami. Up until becoming apart of team Avatar, she has barely suffered any negative events in her life. The one event being that her mother was killed by the triad. Now, that is devastating but that happened a long time ago, however, it didn't affect how she lived. She lived a pampered life, she got anything she wanted and she always had a home plus one loving parent. Had those 'tragic' events not happened to her, she would in my mind be a bland and boring character. Let's try comparing her to Mako, now Mako lived as an orphan for more than half his life and witnessed the murder of both his parents, was forced to become an adult at a very young age and lived on the streets. He never had time to be a kid or a teenager. Asami losing her father is a tragedy but she decided to fight against him which developed her character, and made her more interesting, it was the one event in her life which made her an even stronger character.
Wow, Asami lost her beautiful mansion and doesn't have a gold plated toilet any more. Boo hoo.
Right after Asami left her house, she was offered a place to stay. She still had the comfort of her friends which gave her security, had she not felt safe with them, she wouldn't have joined Team Avatar or stayed on Air Temple Island. So no, she didn't lose every sense of security.
Yes, Asami lost her boyfriend but she knew about Korra's crush on Mako and saw that Mako deeply cared for Korra. I assume that she thought that they liked each other and knew that her relationship with Mako was falling. I don't understand why but why does Asami need a boyfriend to make her feel safe? I mean she can easily fend for herself plus she has friends to make her feel safe.
Korra was thinking rationally. She did the right thing by confronting Tarrlok in a peaceful manner, and had before already told Tarrlok to leave everyone alone. Korra never thought that Tarrlok had gone mad. Yes, it was stupid of Korra to challenge Amon herself but it showed that Korra has major flaws which is being impulsive and hot headed. Those flaws nearly got her bending taken away, had Amon not decided that he would save her for last. That event gave Korra character development in which she finally felt scared and had no idea what to do.
There were many examples shown where Korra was working with her friends. In episode 8 where they were chasing after Equalists, everyone contributed to that fight, not just Korra. In episode 10 where they were fighting against the Mecha tanks, it wasn't just Korra. There are many other examples, so this shows that Korra does accept help from others.
There are more examples of Mako comforting Asami. After Tarrlok had knocked everyone unconscious, Mako helped Asami up. After Asami realized that she couldn't live at her own home anymore, Mako comforted her.
By that time, Asami had already grown up. She never had to grow up by herself, she had her father.
Mako is a great example of how someone might grow up without parents. Mako has shown to be extremely protective of those dear to him. Which is believable because he had to take the role as an adult at a young age. Furthermore, he is inexperienced when it comes to handling relationships. No one really taught him about dating and didn't get to experience the teenage life.
Mako doesn't really get frustrated at others. Though there was a time when he got mad at Bolin for making a deal with the Triple Threat Triad.
I got plenty more examples to show that Mako is not a terrible character, want to hear them?
Again, it wasn't the right time, nor the right place for a proper break up. It was possible for the break up to have happened off screen and just wasn't shown.
I'm sorry what? Korra lost absolutely nothing BUT her bending? Seriously, Korra losing her bending is the worst that it can be. If she permanently lost her bending (which is impossible) then she wouldn't be the Avatar anymore. Korra spent nearly her entire life practising and exercising her bending, if it were to suddenly disappear, then all those years she worked on her bending would be lost. All for nothing.
Tl;dr
