The full word is is pronounced with a /z/, not with an /s/. If this word is contracted with a preceding word, then if the last sound in that word is unvoiced then 's will be pronounced /s/. Otherwise it will be pronounced /z/.

The Original Poster is correct that if a following pronoun begins with H, then if the word is not stressed, there will usually be no /h/ sound:

• Where is he - /'weər 'ɪz i/
• Where's his sister - /'weəz ɪz 'sɪstə/
• Who's his friend - /'hu:z ɪz 'frend/
• What's his name - /'wɒts ɪz 'neɪm/

Notice that in the last example the word is is contracted with the word what. Because the /t/ at the end of the word what is voiceless, the 's is pronounced /s/, not /z/.

Lastly notice that if a pronoun beginning with an H is stressed, the /h/ will be pronounced:

• It's him - /ɪts 'hɪm/

In the sentence above the word him is stressed, so we pronounce it with an /h/. If we see an accusative pronoun like him after the verb BE, it is very likely to be stressed.

^{Transcription note}

^{I have used British English transcription here. The points about /z/, /s/ and /h/ are the same for standard British and American Englishes.}

I recall at university there was an (achingly politically correct) S/he Society, whose members prounounced it Shehe. But outside such specific contexts, you will always have to decide which pronoun you wish to use for a singular person of unspecified sex: he, he or she, singular they? S/he is a written version of the second, and as such is to most people vaguely unsatisfactory but not obnoxious.