Nope. Immunity to mind-affecting effects in case of constructs is part of construct type traits.
Do all constructs lose their immunity to mind-affecting effects if they have an Intelligence score? I assume yes.
Depends on how they gain the intelligence score. Most of the ways they gain a score usually do remove the immunity IIRC but a construct who is stat'ed out with an intelligence wouldn't lose it unless it states it does.
They keep the immunity unless otherwise stated. I have seen two constructs that gained intelligence after creation in AP's. They kept the immunity.
I think there is one construct that is intelligent, but I can't remember the name. I don't think it loses its immunity either.
Bestiary 1 has Retriever. I think that robots from Inner Sea Bestiary have Intelligence as well. All have "construct traits" in their defensive abilities line listed without mention of losing immunity to mind-affecting abilities.