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

1. less是形容词比较级,any可以休是比较级.
2. with 介词短语是multiply(不及物动词,繁殖下去之意)的伴随状语, 与...之意.翻译: 在澳大利亚和新西兰,这个兔子没有天敌,与那些纵欲无制的各色兔子们交配繁殖后代.
3. 你的解释合理.没有on就没有了"接续"下去之意.
4. that 后是teach的宾语从句,宾语从句中又包含一个主语从句.也就是说: whatever ......on a hearth-rug 是 leaves的主语. Whatever 是have 宾语,原型该是: ...that a fellow-guest may have whatever of ..., 但whatever必须位于句首.