Trump will be inaugurated.
To address the OP directly, I don't see how there are 37 Republican electors ready to switch against Trump. Even if some of these electors were Cruz people in states he did well in, I just don't see it happening.
There may be ethics issues within his conflicts of interest down the road, but I don't see that escalating into impeachable territory until something actually comes out that proves wrongdoing occurred.
I think we should expect a tumultuous administration because nothing about him or many of the choices he's made regarding his cabinet/advisers is normal.