Two Questions (Spoilers)

Watched this movie again tonight. Too underrated. Noticed a few things I've never noticed before.
First, how did Roger manage to kill the mom and get away with it? Wouldn't police have been all over her death? She was clearly murdered and if he hid the body then she would've been proclaimed as missing. Especially cause she was a broadway actress on the rise, it just seems like he wouldn't been able to get away with that. But maybe I'm over thinking it.
Second, did it ever explain how they replaced the director that got murdered? He said several times throughout the movie that he was playing one of the male leads in the musical, but then he got murdered a day before the show. It didn't explain how they got around his part. Would they just cut it out of the show? I know it doesn't matter, but the film doesn't even address how they got past that.
All in all though, wonderful movie. Seriously needs more attention.