弁償　Repayment for damages, losses or expenses. ex) 割った窓ガラスを弁償する。　弁償するために～支払う。

代償　Offering a thing, money and goods. ex) 命を代償として、勝利を得る。 迷惑をかけた代償は大きい。

代償 is much more likely to make sacrifices for something.

It is not easy to see the differences between them. You may use 弁償　when it is affordable repayment, such as windows, car or something (Car is expensive, but you can buy it, not a life.)

I then never say that 窓ガラスを代償する。 We do not put verb after 代償, but we put particles, such as として, は, and を after it.

根気　Determination, energy, endurance. 根気強い、根気が続かない、根気負けする。

根性　Innate disposition, mind or nature. 根性のある人。島国根性。

I would not say 島国根気、but 島国根性. I would say 根性が続かない or 根気が続かない. They are really hard for me! 根気 is more like doing continually (Whey you keep doing something with patient industry, you may say 根気よく～を続ける。)

Meanwhile, when you come up against a difficulty, you may say 根性. 根性 implies that you have a strong spirit to get through something. Also, 根性 is often used negatively. ひねくれた根性、根性の腐った like that. I just memorise such sentences to use similar words. 

Maiさん、ありがとうございます。