1 First the electroencephalograph (which is simply a device for recording the electrical activity of the brain by attaching electrodes to the scalp) shows that while there is a change in the pattern of activity during sleep, there is no evidence THAT the total amount of activity is ANY LESS.
这里的THAT引导同位语从句，后面的ANY是ADJ，LESS是NOUN，LESS：SMALLER AMOUNT ，QUALITITIES ETC。
2 This rabbit had no natural enemies in the Antipodes, so that it multiplied WITH that promiscuous abandon characteristic of rabbits.
WITH前置了，我的翻译是：因此凭借杂乱放纵增加了兔子的特性。但总没有人告诉我WITH为什么前置。
3 Later it was found that there was a type of mosquito which acted as the carrier of this disease and passed it ON TO the rabbits.
这里的ON表示“持续”，去掉就不能表示 “持续”。
4 Experience teaches me that whatever a fellow-guest may HAVE OF power to instruct or to amuse when he is sitting in a chair, or standing on a hearth-rug, quickly leaves him when he takes one out for a walk.
我感觉“whatever a fellow-guest may have ”是一个整体，加上of power,意思就是能力上FELLOW-GUEST所具有的，但是资料上解释是：“whatever 是主语(后边可以用of 结构) power:ability to instruct:to teach a fellow guest（共同赴宴的人） may have 做谓语 fellow worker/actor 同事/演员搭挡”，这个解释我看不懂，尤其是“MAY HAVE 作谓语”

以上是我的请教和分析，欢迎大家指点，讨论，谢谢！

你好多多，最近学校工作太忙了，所以虽然看到了你的问题，但是直到今天才有时间坐下来认真地看你的问题，抱歉啊，希望我的回答能够给你一些帮助。
我不太清楚这个句子的来源，但是我认为正确的语句应该是：

1 First， the electroencephalograph，which is simply a device for recording the electrical activity of the brain by attaching electrodes to the scalp， shows that while there is a change in the pattern of activity during sleep, there is no evidence that the total amount of activity is any less.

这里that the total amount of activity is any less 是evidence 的定语从句表示: 没有证据显示在活动总量上有任何得减少

全文可翻译为：首先，通过附着在头皮上的电极来记录大脑的活动电波的简单装置脑电图仪显示，在睡眠期间（依然）有活动状态的变化，没有证据显示在活动总量上有任何的减少

2 This rabbit had no natural enemies in the Antipodes, so that it multiplied WITH that promiscuous abandon characteristic of rabbits.
with that 是个固定搭配，表示接着就，随即，这里multiplied 是动词，繁殖，那么随后的句子应该是表示繁殖的状态或者环境，
在澳大利亚和新西兰兔子没有天敌，因此兔子可以在放任和混乱的特征下繁殖。

正确的句子:

3 Later， it was found that there was a type of mosquito ,which acted as the carrier of this disease and passed, it ON TO the rabbits.
不知前后文，所以只能猜测，ON 表示在---之上，依附于， 因此全句可以理解为
随后，人们发现但当携带和传播这种疾病的蚊子，依附于兔子（‘的身上’或者‘生存’）。

4 Experience teaches me that whatever a fellow-guest may have of power to instruct or to amuse when he is sitting in a chair, or standing on a hearth-rug, quickly leaves him when he takes one out for a walk.

关于这个句子的翻译何理解，我也有点困惑，但是在看了孤鹰远影的解释后，我认为他的理解是正确.
句子的本意应该是想表达

应验告诉我无论这些客人如何具有魅力，不管是他坐在椅子上，还是站在地毯上都能能够给人以启迪和愉悦， 一旦他带人出去散步时，这种魅力就立即消失了。

只有如此的翻译，似乎整个句子才通。所以，这里的quickly leaves hime 应该是指的 power.

Experience teaches me that whatever a fellow-guest may have of power to instruct or to amuse when he is sitting in a chair, or standing on a hearth-rug, quickly leaves him when he takes one out for a walk.

定于从句 whatever a fellow-guest may have of power to instruct or to amuse when he is sitting in a chair, or standing on a hearth-rug, quickly leaves him when he takes one out for a walk.

whatever 在这里应该是形容词，修饰 a fellow-guest，
a fellow-guest 作句子的主语，
may have 句子的谓语，
power 为宾语。to instruct or to amuse 修饰power
when he is sitting in a chair, or standing on a hearth-rug, 作句子的状语从句，
power 同时是后半边句子的主语，
（Power）（主语）quickly leaves (谓语）him when(状语从句） he takes one out for a walk.


写到这里，我自己都糊涂了， 这里的中学生学习，老师从不这么分析句子，因此，中国的学生如果论考试，语法总是很好，可是说出的话，写出的文章却很难懂。我自己现在依然在补习英语。以上的分析仅供参考。