The polite form is to say he is repeating the year. For example, "Peter is repeating Grade 4". "Flunked" is too strong a word as it is very critical. You would never use it for a young student. For someone older, you could use it but it is derogatory so you'd only use it as an insult, joke, or to mock the person. So "Repeating" is the best word word to use, as it is suitable for any age student - primary school through to university. "He repeated year 3 at school" "She's repeating her year at university"

The proper grammar for your questions are... What is the word for.... - instead of How do you say it? And What do you say when a student does not pass to the next year in school? Also - the word choice you used is correct - Flunked. You can also simply say “the student did not pass.”

Repeated or repeating the 3rd grade or held back/held back a year/held back in the 3rd grade are better. Flunked is a harsh word.

I usually say "Johnny was held back in the 3rd grade" or "He had to repeat the 3rd grade." I would use "flunked" or "failed" to refer to a specific class or assignment. "He flunked his biology test" or "He failed English class."