Japanese language/culture questions (best answer will be chosen!)?

I hope you wouldn't mind answering a couple of questions for me. My friend and I are collaborating on an original manga (she asked me to try translating it into Japanese for her, just for kicks because it's set in Japan) and we had some questions. I'm currently studying Japanese at university, but I don't know these quite yet. :/ Here's a basic overview of the plot:

-The main character is fifteen years old

-She has a childhood friend that she likes

-She can't ask him out because she doesn't want him to discover her secret (we haven't figured out the secret yet, but my friend says she's got some ideas)

-Her father doesn't approve of her going out because of her secret (she doesn't have a mother)

-She runs away from home (we're thinking she runs into her mother and her mother convinces her. Her mother and father separated right after she was born and she was raise by her father. She spends the night at her mother's place and goes back the next day)

-Because of this, she misses out on a day of school, but she tells her secret to the guy (Kazuhiro-kun) and gets a happy ending

1. How would you say "don't keep secrets from me?" (this is the main character's father talking to her) I thought it would be "Papa ni himitsu wo kakusanai de." but I'm not really sure; that sounds kind of weird.

2. "Kazuhiro-kun to tsukiaihajimeta kara ikkagetsu gurai tatta." - This is how I translated "It had been about a month since Kazuhiro-kun and I started dating." Is this correct?? (Kazuhiro-kun is the name of the guy the main character likes)

Also, a question about Japanese culture - in the manga, the character misses a day of school, and the way we wrote it was that she gets scolded by her teacher for being absent without permission, but does this often happen in Japanese schools? We thought it would be a good interlude of sorts to show the scene where she makes up with her dad and then her getting scolded in the teachers' lounge, then she seeks Kazuhiro-kun as she leaves the teachers' lounge, and she talks to him. Is it reasonable to have this happen in terms of Japanese culture? Do teachers often scold their students for missing school?