Problem here is, the house can elect anyone they wish, they would not have to go with Cruz if he is the republican nominee. If the election is thrown into the house, and the house is controlled by republicans, then it is likely the establishment house will choose establishment candidate Jeb Bush.
Yes, they can. If Hillary gets say 40% of the vote and Trump plus Whomever split 60% of the vote, they are well within the numbers to say that 60% of America voted against Hillary. A compromise candidate could be in order. Assuming a Republican House doesn’t get browbeat by the media into picking Hillary because she had 40% — something that Boehner is not immune to — I would expect a compromise candidate to be a republican. It would be hard to pick someone who’d already been beaten in the primaries.
They could go after a female or minority and calm lots of dissension. It would probably be a governor or senator.