一，在下列两种情况下，独立主格结构中的being（或having been）不能省略：
a. 独立主格的逻辑主语是代词时。如：
It being Monday, we went to school.
b. 在There being＋名词的结构中。如：
There being no bus, we had to go home on foot.

那么以下两句是不是可以这样改写：
1, The question being settled, we went home.
改为： The question settled, we went home.
还原成状语从句是不是：After the question had being settled, we went home.
那是不是也可以改成：The question having been settled, we went home.
或者把这句理解成原因状语，改成 Since the question was settled, we went home.

2, The monitor being ill, we’d better put the meeting off.
The monitor ill, we’d better put the meeting off.

二，在“名词（或代词）＋介词短语”构成的独立主格结构中，一般不用形容词性物主代词和冠词。如：
Miss Smith entered the classroom, dictionary under arm.
(比较with的复合结构：Miss Smith entered the classroom, with a book in his hand. )

那以下句子：
1，Glasses in his hand, he asked where his glasses were. 是不是错误的？应该是：Glasses in hand？
2，A smile on her face, she asked what she could do for us. 应该是：Smile on face 或 With a smile on her face?
3，He was waiting, his eyes on her back. 这句需要说" with his eyes on her back" 吗？

一、
The question being settled, we went home.是错误的。

因为settle与go这两个动作不可能同时发生、进行，而是有着明显的先后顺序。

所以正确的说法是：The question settled, we went home.。

还原成状语从句应该是：After the question had been settled, we went home.（being改成been）。

另外，The question having been settled, we went home. 的说法语法上是没有错误的，但用现在分词的完成被动形式显得多余而没有必要。当然你要把此处的独立主格结构理解成原因状语，只要逻辑说得通，当然是可以的。

The monitor ill, we’d better put the meeting off. 完全符合语法规则。

二、
Glasses in his hand, he asked where his glasses were. 是不是错误的？应该是：Glasses in hand？
答：是的。

A smile on her face, she asked what she could do for us. 应该是：Smile on face 或 With a smile on her face?
答：yes。

He was waiting, his eyes on her back. 这句需要说" with his eyes on her back" 吗？
答：是的。因为用eyes on back是不合适的，没有表达清楚谁的眼睛看谁的背部。