Although Dan Brown is a very entertaining writer, I often find aspects of his plotline and characters to be highly unbelievable, and rather convenient for the story that he is trying to tell.
Yes, Mal'akh did "win over" Katherine quite quickly; part of this is that she knew her brother was gone, but didn't know that it was quite an emergency, or what was going on. So, her level of alarm wasn't as high as ours was, as knowing audience members. She just wanted to know where he was and that he was okay. All of that being said, I wouldn't go do some strange doctor's house alone at night. I just wouldn't do it. It's not professional, it's unsafe, and highly bizarre for a doctor to ask you to do. She was stupid, in my opinion, to trust her safety to a stranger in his highly isolated house. Peter having mastered the iPhone is more believable; he is an intelligent human being after all, and new gadgets are fun to mess with.
As far as Mal'akh infiltrating the Masons to the highest level and getting all of their deep, dark secrets, that is once again a convenient tool for Dan Brown to use to intrigue the reader to keep going in the book. If we think that he--despite his weirdness, total lack of a background, and violent, malicious tendencies--did somehow get "THE" secret, we will be dying to know what it was. Who cares if it's believable as long as it keeps people going until the end of the book, right? T.V. does this all of the time--they have that long, dramatic pause..."And the next winner of American Idol is.......to be revealed after the next commercial break!" It's a suspense strategy.
For me, Brown's books are less about believability and more about fantastical adventures in which you have to suspend your disbelief to enjoy. There is no way you can enjoy his books if you are analyzing every detail for consistency, probability or even realism. Just let it go, and enjoy the ride--that's how Brown needs to be read.