3. Britain was the first country in the world to have a free health care system paid for by the government.
英国是世界上第一个拥有由政府负担费用的免费医疗保健系统的国家。
(1)这个句子的重点是the first...to do “第一个干……的”,动词不定式作定语,相当于一个定语从句。句中to have 可以转换为that / which has。
As we all know, Wang Zhizhi is the first player in Asia to join the NBA.
众所周知，王治郅是到NBA打球的亚洲第一人。
He was the last to be there.
他是最后一个到那儿的。
(2) paid for by the government 是过去分词短语作后置定语,相当于一个定语从句which was paid for by the government,修饰先行词system。
The student dressed in white is my daughter.
=The student who is dressed in white is my daughter.
穿白色衣服的学生是我的女儿
(3) pay for 意为“付款”，“付出代价”。
He was in such a hurry that he left quickly without paying for the vegetables.
他太匆忙了，没付买蔬菜的钱就匆匆离开了。
You can't be too careful, or you'll pay for it.
你越小心越好,否则你会为此付出代价的。

4. The problem with this system is that poorer people don't have the money to pay for private health insurance.
这种制度所存在的问题是穷人没有钱缴纳个人健康保险。
该句是复合句。that在此引导表语从句,在句中无意义，但不能省略。that除了引导表语从句之外，还可以引导主语从句、同位语从句和宾语从句,在宾语从句中可以省略，其他从句中都不可以省略。
The fact is that he has not been seen recently. (that引导表语从句)
事实是近来谁也没有见过他。
John said that he was leaving for London on Wednesday.(that引导宾语从句，可省)
约翰说他星期三要到伦敦去。
The fact that he has not been seen recently disturbs everyone in his office.(that引导同位语从句)
近来谁也没有见过他,这一情况令办公室所有的人不安。
It is quite clear that the whole project is doomed to failure.(that引导主语从句)
很清楚，整个计划注定要失败。