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Eating a hearty breakfast, we prepared for our long journey.

Why is the underlined part of the sentence a participle phrase?

'eating' is a non-finite verb, not a gerund, but the whole phrase may be considered as a verbal noun (gerund phrase), not a verbal adjective (participle phrase).


All the guests having arrived, the host started the party.
"All the guests" Is it the subject of "having arrived"? "having arrived" is a participle phrase and an adjectival phrase I think

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Being aware of my glaring inadequacy, I tried to work very hard.

"Being aware" I think it is a participle phrase and an adjective. Of genitive is a preposition and "my inadequacy" is is object of the preposition.
"Being aware of my glaring inadequacy" is also a participle phrase, if I am not mistaken.

what is "glaring's" lexical category?
I think "inadequacy" is the object of the possessive pronoun in singular first person, and glaring is the adjectival modifier of the object.
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Being aware of my inadequacy being glaring, I tried to work hard.
Here, being glaring is the objective complement, is it? Also, It is a participle. Glaring is an disjunct and has a different syntactical role from that one when it precedes inadequacy.
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Honestly, the recent measures introduced by the the local government in Myseria seems too harsh, with poor people paying taxes almost more than their income.

"honestly" is an adverbial, and it is also a disjunct, expressing the writer's attitude.
"the recent measures" subject
"recent" and "Myseria" set the scene of the action. So they are adjuncts.
"introduced by the the local government" participle phrase and 'introduced' is a participle.
"
in Myseria" is it an adjectival complement besides being a prep phrase?
"seems" is a linking verb (copula)
what is "too"?
"with poor people paying taxes almost more than their income" adverbial, prep phrase.
"almost more than their income" objective complement?

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Comments  (Page 2) 
Milky<Hi Milky,

"What do you see as the verb for the subject there?" Prepared and tripped.
I hope this time we agree>


If you add "fell", we agree.

NB

<Here, being glaring is the objective complement, isn't it? >
We agree

isn't it = 'is it not' even in tag questions
"is it" is correct too according to my grammar book, but has stronger rethorical effect than "is it not"
<isn't it = 'is it not' even in tag questions>

Not always. Why should it be?

<"is it" is correct too according to my grammar book, but has stronger rethorical effect than "is it not">

It has the effect of "so you are saying that..., are you". Here, nobody said that it was that way before you mentioned it. So you need to ask a question and not make a statement, in this case. Use the negative tag.

Here, being glaring is the objective complement, isn't it? = So you are saying that being glaring is the objective complement, are you?

That can sound rather rude.

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'isn't it' = 'is it not'
"Not always. Why should it be?"
'isn't it' is the contraction form of 'is it not' and not of 'is not it' which is ungrammatical.
Here, being glaring is the objective complement, isn't it? = So you are saying that being glaring is the objective complement, are you?

"That can sound rather rude."

Declarative sentences (it is an adjective) with a positive tag (is it) express emotions (query, worry, anger, surprise)

http://www.englishclub.com/grammar/verbs-questions-tag.htm
I don't know if you really ever listen to any native who wants to help you. A query can also sound rude, can't it?
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