What is the Republicans' view of Trump and how do they plan to use him?
You're right: The Republicans -- especially McConnell and Ryan -- regard Trump as a "useful idiot" who will sign whatever they put on his desk. Trump is proving that their confidence is well placed as he pushes a healthcare bill that breaks every commitment he made about healthcare during the campaign and the transition. However, neither McConnell nor Ryan makes much effort to disguise their contempt for Trump, even as they (Ryan especially) fawn over him to get the healthcare farce passed. It is not hard to see that at some point, Trump will no longer be useful to them; he'll merely be an idiot. They'll turn on him in a New York minute and gleefully embrace President Pence. Trump's incompetence, corruption and lunacy are not sustainable.
I think American politics have already rendered Trump no longer useful to Putin; to him, Trump is no loner useful; he's merely an idiot -- except to the extent that Trump's presidency is destabilizing both domestically and on the world's stage.