1 - They do, but I don't think they go into the water on the current version, and neither have anything programmed to make them take damage there.
2 - Storywise, she can heal anything. Kalyskah is immortal and even if you cut off her head, she'll just reform somewhere like on the last coffin she rested on. We just don't have dismemberment as a mechanic on the game xD
3 - Yep, on the cutscenes that she takes people hearts she actually recovers blood points by drinking from them: 
4 - You can't pet them, but that's actually a good idea!
Once someone invites her in, they can't uninvite her. It's too late, if she is there to harm them they are already lost!
On the game, she bounces out of the doors that she's not invited in, but if you want to know about the lore, getting inside houses without an invitation would make she bleed out and feel excruciating pain, like on this movie: