show (up), help (out), be closed (down)

Show has quite a few meanings, depending on whether it's used as a verb or noun. I suspect you're looking for the most common meaning which would be to let someone see something. As in " I'll show you my pet hamster."

To show someone up is to ridicule them, or point out their failings, usually publically. It has a very specific meaning.

Helping someone out is also very specific and implies quite a lot of help and quite a lot of hardship. You would help out a homeless person or a victim of a crime. Just helping on its own is not as strong. Helping someone by holding a ladder for them is not the same as helping them out.

If somewhere is closed down it means they've finished doing business for good. If they're closed it may be because it's outside of opening hours, but if they're closed down then they're finished.

Again, all the phrases you have chosen have specific meanings, yet the individual words can have many meanings depending on context.

Intransitive show up is American for the British turn up, to arrive when expected.

I'd forgotten about that one.

I believe if an American toddler shows out, it's what Brits call showing off

Never heard 'shows out' in America.

Unfortunately, there is no trick to these phrasal verbs. English has a lot of them. They don't make sense. You just have to memorize them. No one can explain why you fill out a form, or put off a doctors appointment, or break up with your girlfriend, or pick up your kids from school. These phrases are just things we use without ever thinking about them.

I didn't know what a "phrasal verb" was until I started helping people learn English. If you ask an American (Or a Brit I imagine) to explain them, they won't really know what you are talking about... you never see the quirks in your native language.

To Hell and Back
By Michael Jackson
(not the rock star)
Not just small kids, it seems...

Okay, now I've heard it. Once.

Be that as it may, an interesting book is in Google Books:

The Two-word Verb: A Dictionary of the Verb-preposition Phrases in American English

https://books.google.co.uk/books?id=YrkDn8HtQNAC&printsec=frontcover&dq=inauthor:%22George+A.+Meyer%22&hl=en&sa=X&ved=0ahUKEwjhm7XU3sbXAhWmCMAKHcfSC9gQ6AEILDAB#v=onepage&q&f=false