***I'm not a teacher***
He/she could come outside to the garden and pat the speaker's back as an example of calling him. 'Call' in this context just means get attention or notify.
1. Yes, through is correct. Judging from the context, you would be inside the house and the speaker will be in the garden. You can see through the window because the window is made out of glass and is transparent so light could go through the window, allowing you to see through it.
2. You can't call someone over the window unless it's open. Giving a shout is fine as well as long as it brings the speaker to your attention.
Student or Learner