I'm definitely not a linguist but I am a native english speaker in London and I'm quite sure I've only ever used "he's = he has" when it has a verb coming after it. If "has" means "belongs to" it definitely doesn't get contracted. I'm pretty sure I've heard "He's himself a dog (for example)" where "He's" = "he has (possessive)". I don't know where this is from I'm guessing very far south in the UK maybe cornwall / maybe a stereotypical "english farmer" speak thing. Maybe it doesn't get contracted and I'm wrong.

At least in some Northern English dialects it wouldn't be uncommon to hear "He's a right way with words that one!" or "They've a car that needs fixing..." or other similar constructions. However, I think it probably has more to do with the dropping of the word 'got' in the present perfect than a contraction of the present tense of 'to have' on its own, at least in most cases. With that being said, it wouldn't suprise me to hear such a contraction, especially within the phrase 'They have a', with it later being pronounced "Theyvuh" (e.g "They've a house up the road."). I agree that it's not something you'd hear often and that it is probably very dialectal but it wouldn't be an impossiblity either.

He's (possessive) is incorrect in this case.

He's can be used as short for "he is" and "he has"

He's not feeling well = He is not feeling well

He's been very ill lately = He has been very ill lately

To me, he's a cold is not entirely correct. It sounds more like you are saying that he himself is a cold and less of he has a cold. In such cases we have to introduce "got" to eliminate that confussion and misunderstanding.

Therefore, he's got a cold would be the correct statement.

I hope I'm making sense.

"he has a cold" = what you expect.

"he's a cold" = "he has a cold" for some British speakers = the same problem as for gonna wanna shoulda etc that many learners take many years to understand and sound ridiculous when actually trying to replicate the sound; discussed frequently on this discussion forum.

"he's a cold" = also it means he is a cold (whatever) "he's a cold son of a female dog"

That's the only over way I have heard it but it looks doubtful this is the case for the situation being discussed, so I assume it is the British accent for many British speakers causing the problem. Even though I am a Londoner often having to endure jokes and bad London speaking imitations I personally always say "he has a cold" taking care to separate the "he" and "has - a".

This has nothing to do with the stereotypical cockney dropping of "h's" that occurs only on some words. The two most common ones being "house" and "hill" "'ouse' and "'ill".

And @Liam Patrick also has a valid explanation of what might be happening for a regional British dialect.

Also I have heard @Mathew Beardwell''s example both in London and in more rural areas

You can also say "he's got a cold" .

Here, the short form is correct, isn't it?