1. While she felt embarrassed, she ran out of the room.
2. Feeling embarrassed, she ran out of the room.
3. Embarrassed, she ran out of the room.
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Are they all the same in meaning? Can we remove 'Feeling' in Sentence 2?
2 and 3 are fine.
In 1, the intro word should be "When" in order to make the same sense as 2 and 3.
Yes, all three sentences have the same meaning. In all three sentences, the subject (she) is feeling embarrassed, and as a result, she ran out of the room.
In sentence 1, the word "while" is used to indicate that the feeling of embarrassment and the action of running out of the room happened at the same time.
In sentence 2, the word "feeling" is used to explicitly state that the feeling of embarrassment motivated her to run out of the room. This word can be removed without changing the meaning of the sentence. So, you can say "Embarrassed, she ran out of the room."
Sentence 3 is the most concise form of the three, where the feeling of embarrassment is described with the participle "Embarrassed" at the beginning of the sentence. This structure is often used to convey a cause-and-effect relationship between the feeling and the action that follows it.
Therefore, you have the flexibility to remove "feeling" in sentence 2 and still convey the same meaning as sentences 1 and 3.