They'd have to move left on economics and deemphasize and/or offer a moderate solution to social/"woke" issues, while caring about the problems that impact minorities. Presenting themselves as a moderate alternative to the Dems like what The GOP did during the '30s/'40s would be the only way to win at this point, barring a devastating administration ala. Truman-style or an extremely popular GOP nominee. I cannot think of a GOP nominee right now who would win based off of bipartisan popularity (not even someone like Romney or Cheney), but I'm sure they'll be someone someday.
If the election was held today I think a few would but they wouldn't after a whole campaign of attacks.
Absolutely, I don't think it would be immediate, but I think after losing the W.H. at least three times, they'll likely moderate. I have a feeling if Trump lost to HRC, they would've reverted back to the 2013 memo and run someone like Hogan or Kasich who while conservative, can appeal to moderates who normally vote Dem. If they still lose in 2020, they'll probably try running someone like Baker or Scott and if they still couldn't win in 2024, they'd look at where they went wrong and re-evaluate by 2028, likely following my suggestion until they can expand their base enough to win and then going from there. Conservatism might return to the party eventually as it did under Reagan, but I wouldn't of counted on it until they became competitive again, which might've taken a number of decades.