The two participle clauses you cite are in the passive voice, and being puts them in the present continuous tense. They are continuous passive participle clauses.
Being beaten by (the*) snow, he died.†
Beaten by (the) snow, he died.
The snow beat him, (and) he died.
One of your opinions about the two sentences is correct: being+past participle is identical in meaning to the past participle alone, and the gerund being can be omitted in both sentences.
Being beaten here is a continuous passive participle. Without being, it is a passive participle. Their meanings are nearly identical, and being can be omitted, but as you suggest, being in the first example emphasizes that he died because the snow beat him. It "tells a story", in a way, of the beating as it progresses. (In the longer and "optional" part of this answer, because is discussed more thoroughly.)
The verb to be is not just a useful and meaningless sound used to form tenses and the passive voice. Be is a "real" verb with a "real" meaning of its own. Being means having place in the objective universe or realm of fact; existing. In your sentence, Being beaten means Existing in the state of "beaten"-ness. Understanding what being really means, and thinking deeply about the verb be in relation to your two sentences, may open up for you an understanding of how the passive voice works in English, and the other ways in which we use the verb.
The verb to be is often taught as a linking verb or auxiliary verb, and it performs those duties very well. However, without knowing what the verb means, it is impossible to truly grasp the sense of many constructions in English which employ be in its various forms—including, I think, its workmanlike service in linking and forming tenses and the passive voice.
In your first example, the verb to be helps to describe the cause of the subject's death:
Being beaten by the snow, he died.
Being beaten is routinely analyzed as a continuous passive participle using the gerund of the verb to be, but with the "meat" of meaning provided by the participle beaten. Analysis might center on that verb's meaning; but the verb be is important here, even though it could be omitted without changing the sense of the sentence:
Beaten by the snow, he died.
What does it mean to "be"?
In most beginning English classes, the verb to be is the first one studied. Because it is the most irregular of verbs, and because of its wide use as a linking verb, the forms of the copula (as it is known to linguists) are memorized early on.
Webster's Unabridged Dictionary (1913) has this to say about to be:
To exist actually, or in the world of fact; to have existence.
To exist in a certain manner or relation, -- whether as a reality or as a product of thought; to exist as the subject of a certain predicate, that is, as having a certain attribute, or as belonging to a certain sort, or as identical with what is specified, -- a word or words for the predicate being annexed; as, to be happy; to be here; to be large, or strong; to be an animal; to be a hero; to be a nonentity; three and two are five; annihilation is the cessation of existence; that is the man.
To take place; to happen; as, the meeting was on Thursday.
To signify; to represent or symbolize; to answer to.
Many words are used to define this very small one, and rightly so. It is a part of the very "framework" of English (and of all Indo-European languages) and is so entwined in our languages that it has even "disappeared" in some of them: it is "understood" rather than spoken! After all, what is more elementary to language than the expression of existence itself? That is exactly what to be means.
When we say that something is, we are expressing its existence. When we say that something was, we express its existence in the past. The participle phrase Being beaten uses be in the sense of the second definition above: it expresses the state of existing with the attribute of "beaten-ness."
In comments, you were asked to compare the verb to eat and its participle eating with the verb to be and its participle being. Of course anyone knows what it means to eat; and anyone knows what the act of eating is. No matter what language they speak, all humans engage in eating. The same is true of being, and the definition of that word is easy to grasp if you consider that:
Being is to be as Eating is to eat.
Of course, eat is a dynamic verb that describes an action, and be is a stative verb that describes a state or condition, but it is easy to understand that being and eating both express what happens when the action or state described by the verb is undertaken or experienced.
Here we can come back to the part of your question in which you suggest that the participle phrase Being beaten replaces the adverb because. You are nearly correct! It is Being which could be replaced, and we can express the same thought like this:
Because (he was) beaten by the snow, he died.
Yet even here, the verb to be is involved: The adverb because is itself descended from the verb be and the noun cause! As Webster's 1913 Unbridged also tells us in its definition of be:
- Note: It is joined with certain substantives, and a few adjectives, to form verbs; as, bedew, befriend, benight, besot; belate (to make late); belittle (to make little). It also occurs in certain nouns, adverbs, and prepositions, often with something of the force of the preposition by, or about; as, belief (believe), behalf, bequest (bequeath); because, before, beneath, beside, between. In some words the original force of be is obscured or lost; as, in become, begin, behave, behoove, belong.
In the original form of because, the words be and cause were separate:
For be-cause þat I know þe sorw þat þay haue, ich am com doune to deliuer hem.
(From a sermon composed in the 1400s by an unknown writer and preserved in a manuscript at Worcester Cathedral.)
& þer power was adraw be cause he made þe signe of þe cros...
(from a Middle English Treatise on the Ten Commandments)
And Geoffrey Chaucer in The Merchant's Tale writes:
If Hys Squyors / whiche fat stoodon ther bisyde
Excused hym / by cause of his siknosse
A discussion of the verb to be cannot but end by noting that it is the subject of what some believe to be the most perfect poetry written in English. In 1600 CE, The Bard of Avon put these words in the mouth of Hamlet, the Prince of Denmark:
To be, or not to be, that is the question:
Both of your example sentences numbered as 1.) contain clauses in different voices and tenses, which presents problems not addressed by your question. In English, they would at least be written to agree in tense, e.g.:
Being beaten by snow, he dies.
Being taken to the hospital, he survives.
* I have here added the definite article which the questioner omitted in his example.