I’d like to add that these are the terms used by the FBI to describe wanted fugitives on their Top 10 Most Wanted list

In my experience you can use all three with 'build' But, you tend to hear the sizes connected with the word build more in the field of police work. I'm referring to suspect descriptions. In America we tend to say someone is either heavy or medium weight or thin. At least that's what I've generally always used. Of course you could be crude and call someone fat. Skinny is the same as thin and isn't rude at all (especially if you're telling a woman that she's skinny).

I'm not sure that we say 'thin build' very often - 'thin' refers more to a lack of body fat. We tend to say 'slim build' or 'slight build'.

Yes. You could use any of these three.

'He has a heavy/medium/thin build' = He is of heavy/medium/thin build > Both ok