Present participles, like "being", act as nouns, adjectives, or adverbs. In this case, as Karen said, it's an adjective that modifies "water". "Being cold" is an adjectival phrase anchored by the adjective "being". Next, let's think about the other adjective, "cold". What is its role? Unlike active verbs, participles do not have a subject (though they can have an "implied subject") They can, however, have objects and predicative complements just as verbs do. Here are some examples: "Singing a song, the boy felt happy" ("song" is the object for "singing", "singing a song" is an adjective phrase that modifies "boy". "Boy" is the subject of the verb "felt". "Singing" is not a verb so it has no subject. It does have "boy" as an "implied" subject since it is the boy who sings. However "singing" does not MAKE him sing. It only DESCRIBES him singing. That is a subtle, but very real difference. Don't ever think of a participle as a verb. "Singing a song" stands on its own as an adjective phrase. It doesn't need anybody to sing for it. For example, you can say "Singing a song is a wonderful thing to do", and that has nothing that could be misinterpreted as a subject. "Feeling happy, the boy sang a song." ("happy" is a predicative complement for "feeling". "Feeling happy" is an adjective phrase that modifies "boy".) In the adjectival phrase "being cold", "cold" is a predicative complement for "being".

the water being so cold = how cold the water is

It functions as an adjectival clause describing the water.

Hi! In this context 'so' means 'to that extent' or 'that much'. In mandarin, I would say 我更担心水那么冷。 The translation in your book of 水很冷 is more for 'very cold', which is also correct.