We know that phonemes are the smallest unit of sound in speech, and that in the IPA, each character represents only one sound. Wouldn't 'air' be considered two sounds - the combination of the sound /ae/ and /r/?
You give three different example words, and the 'air' sound is not identical in each of the three. Please remember that, relative to my answers.
Your headline asks: Why is the sound 'air'...considered a phoneme?
The sound 'air' is not considered to be a phoneme. It is a combination of vowel and consonant sounds. Any combination of vowel sound and consonant sound is not a phoneme, since the vowel and consonant units are separate and distinct.
Your text asks: Wouldn't 'air' be considered two sounds - the combination of the sound [sic] /ae/ and /r/?
No, 'air' is rarely considered to be two sounds—it is typically considered to be three.
In all my sources, 'air' is signified by three IPA sounds when the word's final /r/ is voiced—it is signified as two sounds, when the word's final 'r' is unvoiced.
The sound 'air' is written—in every source I checked—with separate IPA characters representing both of the vowel sounds in its diphthong, and another IPA character representing its consonant sound. For all your example words, in all my sources, the vowel sound appears as the diphthong ɛə. That's two vowels elided into a diphthong—two sounds, in this case two phonemes. The consonant sound is invariably signified as ɹ for American English—one sound, in this case one phoneme.
In some words, the /r/ sound is not voiced. My sources leave the terminal /r/ sound off of their IPA signs that signify the sound for the word pear. My sources do list the /r/ sound (that is, the IPA character ɹ) for your other example words, chair and where. So, in pear the 'air' sound is two phonemes; in chair and where the 'air' sound is three phonemes.