I can't give you a good explanation. I can only give you examples of (US) usage. When I get off the train, it takes me less than five minutes to walk home. When I get off the train, it takes me less than five minutes to walk to my house. I'm leaving now, I should get home by six. I'm leaving now, I should get to my home by six. In the examples without "to," it feels to me as if perhaps "home" is working as a modifier of "walk" rather than as a noun, although it's curious that if follows "walk." We can and do use "home" as an adjective in phrases like "the home team" or "home cooking."

go is an intransitive verb which means that it should be followed by an adverb / or nothing at the end. you say go + home (adv) go + (to + school) adding 'to' + school (noun) converts it to an adverb.

That's a good question! I assume that native speakers just shortened some typical phrases to save time when speaking. In this case, "get" is the same as "arrive" (and we could also say, "I will arrive home in 20 minutes"). You're right—there's no logical or grammatical reason for these, but it is the way 99% of native speakers say them.