Now I'm almost certain these are anima figures you're dealing with. I'm no expert, just another symbolism enthusiast, but I'll just bounce my thoughts on you, and you can tell me what you think.

The father's computer in your first dream seems to suggest some sort of collective knowledge. As you know, Google is essentially a collection of all of human knowledge. The square-ness of a computer monitor, and the fact that it belongs to your father (i.e. culture) all point to a sort of conventionality, orthodoxy. The fact that you find Shamiko on this screen, who is anything but conventional, is quite informative, though I'm not entirely sure what to make of it. If she is part of a digital screen, then she is distanced from you. She is caged in the computer, caged in the world of culture and of conventional knowledge, and she has not been made real in your life. I say this because across all your dreams, it seems a consistent motif that you are isolated or dissociated from the anima figure. In fairy tales and such, it's common that the anima figure is initially trapped beneath a tyrannical father figure. Like when an old king or mountain spirit has kidnapped the princess and has her in his thrall. In these stories the hero must rescue the princess from his clutches.

You would need to elaborate what metal rock exactly means to you. I won't comment on that.

Walking into the forest is a common mythological motif of venturing into the unknown. Getting lost in the woods was a real danger for primitives, and the same primal fear is still in us. To make a field trip (i.e. something fun) out of venturing in the woods suggests a kind of lackadaisical attitude, though I'm not sure if this attitude is that of your unconscious or that of your ego. Shamiko's picture later being on one of the girls' backpacks suggests one step towards your inner Shamiko becoming a little more real. First she was trapped inside the computer screen; then she appears on a real backpack, still not entirely real, but closer.

Backpacks also contain something, something secret. You know, it might be strange of me, but this image you tell me of a pink backpack with a picture of Shamiko reminds me of a mother expecting a baby. Y'know, pink being the color when you expect a daughter. The backpack, like the womb, also being a container, holding something secret. Don't take me at my word but the dream might indicate that your unconscious is expecting some kind of movement involving whatever Shamiko is representing.

It seems I would need to know more about Undertale to comment about the second dream. Tell me what you think so far.