It's very likely that the author didn't define her age at all. She behaves like a child (a diary, also her size related to the chair, also the idea of exploring dreams in sort of innocent way.
She sort of thinks like a child, dreams like a child, has problems like a child - the guy with piano, kids who don't want to play with her, scary monsters chasing her in dreams, fantastic scenery's, childish infatuation related dream elements
Children more often than adults remember their dreams, I had a lot of pretty weird dreams as a child (I think they were more creepy then those I have now), dreams about fantastic worlds, weird underground labyrinths, my city but changed to empty and weird, abstract looking creatures, dark knights, weird jungles and all sort of stuff
Details like age aren't needed to make an interesting story, sometimes you can even drop the gender... and if the story is good it may happen that some people will read through the whole story and not notice the lack of gender or age of the character at all (sometimes it's better like that, it, in fact, can make the character easier to relate to). A blank character can work as well (check main character of chrono trigger or golden sun, they don't say even one line of dialog through the whole game, while the other characters talk quite a lot, and you may play the whole game and not notice it)
I had some pretty weird sexual fantasies as a 14 year old, and girls mature earlier.
A child can kill and can commit suicide, in fact it's plausible because the child may have lack of understanding of the idea of death, or lack of fear (which is created by the society, not instincts), never seen parents having to stop children from doing dangerous things like walking too close to the edge of a cliff, putting a hand in fire, puling a dog's tail? ya, fear of death is not really something which you are born with, on the contrary
As for living alone, Madotsuki doesn't leave her room at all, I don't think that would work
Personally I think I see Madotsuki as something between 8-13 years old