All the verb forms are fine. That's all good. Just a few other things to point out: Each of your examples is actually TWO sentences. The two statements cannot be connected with a comma. You have two simple ways of correcting this error: either splitting each one into two sentences with a full stop/ period, or inserting "and". For example: He likes to eat ice cream. So do I. He likes to eat ice cream, and so do I. As you can see, I've added the missing "to" to the above sentences. The verb "like" has to be followed by a noun, a gerund ( liking ) or "to" and the infinitive. Finally, you might be pleased to hear that the English language doesn't vary as much as you think. There really is no need to add "in the US" to all your questions, thereby excluding the rest of the world. It makes sense to mention one specific English-speaking country if you have a question about lifestyle or culture, or perhaps a slang expression. But when it comes to grammar, there really is only one English language. The grammar of English is virtually identical in all countries. And even where small differences do exist, these are just preferences or tolerance for non-standard variations. Just so you know! :)

All are correct and sound natural. You will often hear them with an "and" in between, for example "He is going to school, and so am I." For some reason, I think this is especially true for the ones that begin with "neither". Another common way to express the idea when it is positive: "He is going to school, and I am too."

Yes, great job. All I saw is that you are missing the word "to" in your sentence about ice cream. "He likes to eat ice cream, so do I."