Re: differences between e.g write and write down
Welcome to Using English.
When you write something down, you are recording it for your memory or to create a record. You write down a recipe someone told you, a phone number, your grocery list, so you have a record of it or so you don't forget it when you need to use it.
You write most things. You write a letter, a report, an essay, an article, a novel, etc.
I wrote a novel. You might want to write down the title so you can find it at the bookstore. -- Do you see the difference?
I don't see much difference between "chop" and "chop up" except that "chop up" is a bit more emphatic. Perhaps the entire quantity is chopped when you chop it up.
I'm not a teacher, but I write for a living. Please don't ask me about 2nd conditionals, but I'm a safe bet for what reads well in (American) English.