问题1: 问题中It is recounted of中的of 是什么作用，of的常见含义是“属于……”，如：

the back of the house 房子的后部。

也就是说这个后部属于这个房子的，于是可以联想这个后部与这个房子有关。因此，问题中的of 在本句中做与……有关讲。recounted有讲述，叙述的含义，那么它讲述，叙述的是什么呢？是that when he heard of the illness of his friend, Henry Taylor, he went off immediately to visit him, carrying with him in his pocket what remained of a bottle of medicine formerly prescribed for an indisposition of Mrs. Carlyle’s. 由that引导的主语从句就是讲述、叙述的内容，而这个内容与Thomas Carlyle有关，说白了就是讲的是发生在Thomas Carlyle身上一件事。整个句子意思大致是：

People told a story about Thomas Carlyle. Once he heard one of his friends, Henry Taylor, was ill, he went off immediately to visit him. Mr Carlyle wife had been ill and taken the medicine. Now there was some left in a bottle of medicine. So it was strange he had them for Henry Taylor.

问题2:第二个黑体of又是什么作用,of 在本质还是当属于讲。解析上说：of a bottle of medicine ... of Mrs. Carlyle’s作 what 的定语是对的。在这里的what含义是the thing which,而of a bottle of medicine ... of Mrs. Carlyle’s修饰the thing，然后再用which引导一个定语从句加以评论，修饰。这样，the thing被两个定语修饰，一个是of a bottle of medicine formerly prescribed for an indisposition of Mrs. Carlyle’s.，第二个被which  remained修饰。 

问题3: 这个是名词所属格是英文双重属格用法，这里的of含有全体与部分的关系。the back of the house 房子的后部，后部属于房子，后部是部分，房子是整体。比如：

He is a friend of my father's friends. 我父亲有很多朋友，他只是我父亲众多当中一个，部分与整体关系，英语习惯上把friends省掉，于是就有He is a friend of my father's。这方面可以阅读有关语法书。