英语语法: 在修饰分词 “being” 的用法
Source: Zhihu
(prepared by: alexcwlin; edited by: Adam Lam)

Question #1:
我发的第一种，我不明白being done什么时候表示单纯被动完成和什么时候表示被动进行，所以我发的第一个信息里面的being不知是否可以省略

然后我发的第二种，表示一种状态或者后面加形容词(包括分词形式形容词)，这种我看有省略的，我不明白他们省略或者不省略有什么区别。
(The reader then went on to give me a number of sample sentences to demonstrate his bewilderment).

我读了很多语法书，发现没有一本语法书过多涉猎这个问题
(alexcwlin: The comments below are based on my grammar knowledge. Let me know if you have further questions.)

alexcwlin comments:

1.     Here we are talking about Participle Modifiers (修饰分词), with “being done” as Present Participle, and “done” as Past Participle.

2.     “Participles” are Verb forms of modifiers, and so there is action involved in each of those participles.
For Present Participle, the Participle action is/was ongoing relative to the context of the sentence.
For Past Participle, the Participle action has/had been completed relative to the context of the sentence.
Therefore “ongoing” or “not ongoing” will determine if Present Participle “being” should be used.
(Note 1)(Note 2)

3.     Mainly your sample sentences have nothing to do with Passive Voice, and so we leave that alone for now.

Let’s go over your sample sentences:

A.    Present Participle Sentences

a.      The rate of heat production depends also in the power being used in the heating element.
(Note: Participle “used” is ongoing as the context “rate of heat production”).

b.     Being surrounded by mountains, the village looks very beautiful.
(Note: “surrounded” and “looks beautiful” were concurrent events at the time.)

c.      Being beaten by his father, the boy screamed loudly.
(Note: “beaten” and “screamed” were concurrent events at the time.)

d.     Being watched by so many people, she felt very nervous.

(Note: “watched” and “felt nervous” were concurrent events at the time.)

e.      Being tired after a long walk, he sat down for a rest.
(Note: “tired” and “sat down” were concurrent events at the time.)

B.    Past Participle Sentences

a.      The man kept his eyes ____ on his baby for quite some time. A fixed B being fixed
(Note 2)

b.     Heated, water changes into steam.
(Note: “heated” was completed before “changes into steam”.
You can change it into Present Participle like this: “Being heated, the water in the pot was changing into steam.”

c.      Frustrated, he went back to his home town.
(Note: “frustrated”, and then he “went back”. You could use Present Participle like this: “Being frustrated, he is planning to go back to his home town”.)

Footnotes:

1.     See my grammar notes on “Participle” at: https://zhuanlan.zhihu.com/p/44852039

2.     Some Verbs, such as “a bomb exploded”, “a man killed” and “eyes fixed on” often do not make sense if Present Participle is used. In this case, “fixed” meaning “directing something” would not make sense if you say “kept his eyes being fixed on his baby” (because logically “being fixed” is “not fixed”) instead of “kept his eyes fixed on his baby”.

However, “fixed” could have meaningful Present Participle if it means “repair”. For example, “Being fixed, I will not have use of my laptop for the next week.”

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Question #2
Being beaten by his father, the boy screamed loudly. 这个句子我应该是看懂了，强调同时和进行，他爸爸正在打他，他正在哭(还没打完，哭和打都在进行) Being watched by so many people, she felt very nervous. Being Surrounded by mountains, the village looks very beautiful. 按照给我的解释，那么这两个也是同时发生得事件，但是如果我去掉being Watched by so many people, she felt very nervous. Surrounded by mountains, the village looks very beautiful. 这个样子的话，可以吗？同时句子想表示和加上being有什么不同呢？

alexcwlin comments:
“Surrounded by mountains, the village looks very beautiful” is fine because the “surrounded” action can be seen as having happened already.
I would say “Watched by so many people, she felt very nervous” is not OK because “watch” was an ongoing action and if people watched her in the past and not as it happened, she shouldn’t be nervous.