We find out in the movie that Shaw was sent to the Hospital to kill Letty on Braga's orders after Braga learns that Letty survived the encounter. Shaw knew where Letty came from (driving for Braga) and her skills (if she didn't have mad skills, she wouldn't be driving for Braga in the first place), so when he gets to the hospital and finds out she has amnesia, he considers her a blank slate (as he stated in the movie). He figures due to this, he can mold her anyway he wants. He wasn't going to kill her because he believed he could remake her in his own image. If she ever did recover her memories, he could kill her then if she went against him.
Shaw knew Letty was originally a plant from the FBI, but didn't know she was connected to Dom until the picture is shown of Letty sitting on Dom's lap. He did know about Brian, Dom, and the crew due to Riley working on the inside for him, but not about a connection between Dom and Letty. He gets the names of Dom and the crew from Riley and is not overly worried about them until they almost foil the plan, then at that time he gets the in-depth on them and discovers the connection between Dom and Letty. Shaw tests Letty with the cross and picture to see if there is going to be an issue, but is relieved when Letty shows no knowledge of the connection. His soldier is still in place without question.
I agree with you that kidnapping Mia was not part of the original plan. Shaw realizes he needs an ace-in-the-hole if he is going to beat Dom after their encounter (right after Dom and Letty race). He needs leverage over Dom and views Mia the perfect pawn to help manage his schemes. I am sure at this point he knows of the connection between Brian and Mia as well, so this works out doubly well into his plans. We don't know when he comes up with this plan, but he doesn't decide to execute it until after the encounter because he tries to get Dom to walk away, which Dom is unwilling to do.