Breathing in the first sentence is a participle acting as an adjective.
The second sentence is an auxiliary verb. is and breathing make up the verb.
Breathing is ... go no further. Breathing is the subject of the third sentence. It can only be a noun or a pronoun when it is a subject. So here it is a gerund.
The twelve morphological classifications of structural grammar include adjective, noun, and auxiliary. Transform is not one.
