Can the contraction has 's be added to any noun?

Not that way.  We cannot contract the primary verb except for the basic is or are (he's here, we're home!). We can only contract other words when they are secondary (auxiliaries can be contracted).  The roof's leaking, you could say.  But not the roof's a leak because that would mean "the roof is a leak" based on rule 1 about contracting is and are.  Clearly the roof is not a leak, it has a leak, or it is leaking (it's leaking)

s only when the noun is plural,

 cars

 the car is in bad shape,=,,,, the car's in bad shape

the cars' headlight is broken ......cars' is possessive

my dog is sick      = ,my dog's sick

The contraction of 's is of is. That can only able to be added to singular nouns since the contraction of plural nouns is 're being of are.

No. You can only use "'s" in place of "has" when it's part of the present perfect. "He has been sick lately." "He's been sick lately." In other sentences it's likely to be confused with "'s" used in place of "is."

No, that is wrong.