I thought that all appropriation bills had to originate in the House.
The Senate can. But the House would have to pass something. And the two would have to conference and then pass what comes out of conference.
They do. However, the Senate can take an existing House bill and gut it with new language - even if its completely different from the original bill.
You are correct.
Would appear President Trump was suggesting a 'shell' bill which Turtle could originate in the Senate then pass to the House for funding, then pass back. A rare bird, indeed. IF the House passed such a Senate 'shell' bill verbatim, it would not have to pass through a House Committee. To accomplish this is a fait accompli, literally NO ONE in the Senate or House could raise objection to such a package if the goal is to put it on the President's desk.