Do you have to pick up your luggage and go through immigration when transferring *international to international*?

Yes. Because you have through US customs.

Generally speaking, yes, though some exceptions like traveling from Canada does not require it.

Yes

If you don't clear US Customs in the in the country you are departing yes you will have to. Example, when you depart Aruba or Canada, you clear US Customs in that country before departing.

Yes because you have to go thru customs.