- Half of them is/are smart.
*If half refers to one, and them refers to two.
- My heart and my heart is/are beating.
*If my heart and my heart refer to the same thing.
Neither would occur in natural English.
If there are two objects and you want refer to one you should say
One of them is …
If you don’t know how many there are (there could be two or more)
Half of them are…
But don’t use this phrase to refer to exactly one object.
Don’t say “my heart and my heart” if these are the same object. One would probably use “are”, as the only reason to repeat is to make the point that “my heart” actually mean different things (though I can’t imagine how). It is simply incorrect to refer to one heart as “my heart and my heart”.
I think your bigger problem is that you are trying to treat English like a computer language and are trying to understand the exceptional cases in the syntax. For a computer language, there is a strictly defined definition of the language and so it makes sense to ask how (for example)
int i=1; i=++i; would be interpreted
It doesn’t make sense to ask for correct grammar in unnatural expressions in a natural language. Humans are not computers.