For clarification: שם משפחה is construct state because משפחה is a noun so שם does not have ה in the definite. פרטי is an adjective however so it is השם הפרטי in the definite. Can someone confirm this for me?
Yes, you are correct.
MAhu shem ha mishpaKHA shelaKHEM
Shouldn't שם המשפחה be "family name" instead of just "name" ?
It'd "last name" (which is the name the whole family shares - which is the "family name")
If only שם is used, would that imply a first or last name in Israel?