I believe you need to understand, sympathize, even love your characters to write them sincerely. This is especially important, and hard when it comes to antagonists. But in order to make them feel three-dimensional, you're going to be forced to try to understand where they come from.
On authors mental health. This makes me think of August Strindberg, who had an "inferno crisis" where he wrote letters in which he talked about being followed by a secret cabal. Later analysis of his letters has also suggested that rather than having mental health problems, he might have played with scenarios and even written the letters "in character." He did finally publish a book, Inferno, with similar events. Did he suffer from mental health issues or was he taking his research to an extreme level?
Not to mention that every author of a murder mystery has been planning how to kill someone... and get away with it... if you didn't know they were working on a novel...
About being in character. For an author, this is essential when writing dialog, describing actions, and writing POV narration. Adding even more complexity, you need to jump between different characterizations when writing dialog and actions, even though the character talking or acting isn't the POV character.
For falling in love. Yes, why not?
We see a version of this in online dating, especially the chatroom, online world version where people can spend time just being online and falling in love, just to realize, in some cases, they did in fact not fall in love with the person on the other end of the chat, but an, all too perfect, fantasy construction in their own heads.
When you start editing the book on your own, and perhaps, later with a publisher, some of that emotional investment might ebb out as you start getting a bit of a distance to the work. Especially when you realize you have to remove big chunks of the text, even characters you might love... You go from the dreamy world of the first draft to the harsh reality of editing "this thing" into something someone other than your mom might want to read...
If we're talking about readers falling in love with novel characters, then the answer is even more "yes." And, as others have mentioned, when it comes to visual media, even more so. (Who didn't fall in love a bit with Daenerys when she started liberating slaves, just to go "WTF?" when she went "WW II Hamburg" on Kings Landing in the end?)