Grammar – Correct Usage of “I Am Home”

First of all we need more context to comment on that sentence.

We generally use escape from doing something or escape doing something synonymously.

So your quoted sentence sounds like He escaped something to go study

Now coming to your confusion - go study

It's not any elliptical form of go to study, rather it means go and study. It's more common in American informal language, mostly in conversation.

Go shopping, go playing are all different construction, where shopping and playing are gerund.

Hope this helps. Please write back to me if there is any confusion about my answer or if you need any more clarification.

Your use of not using "to" before "home" as it is an adverb is correct, as well as your second statement. That use of "You come to home" is really, really strange and I've never heard that before. It seems to be a typo. I would say "You come home and you want to tell your brother what your friend told you." Either that or "You come to your home...", though the first one is much more natural.

Anyways, to answer your question, here are a couple of SE posts!

https://english.stackexchange.com/questions/33652/why-is-to-not-used-before-home

This one details how the word "home" functions in several ways (how it can be either a noun or an adverb).

Can we use "to" before home, if we are using determiners (her, my, your etc.) before home?

The second explanation is very, very good. It details how "home" acts when bare and surrounded by other things.

I hope this helps, and feel free to ask if you have any more questions or if I didn't answer properly! :)

I'm a native English speaker, from the Midwest (US), if that counts for anything. There are variances in language, so if this is a regional thing, I don't know about it and I apologize for that.