Always!
Can you, or should you? You can hyphenate it if you're moving between lines in a paragraph and need to break up the word. You shouldn't hyphenate it normally.
Yes it should be hyphenated.
Don't hyphenate; ongoing is one word.
If you absolutely must hyphenate a word at the end of a line then you should do so between syllables (and not in the middle of a syllable) to ensure meaning remains intact. A good way to hyphenate this word would be: investi-gation.
You do not generally hyphenate the word stepchild, because it is not a hyphenated word. If the question is where to hyphenate stepchild if you have to break it across two lines, it is not difficult to determine this, because the two syllables are actually separate words: step and child. If you have to hyphenate the word to break it, you would put step- on one line and child on the next. If the word is all on one line, you do not hyphenate it.
If possible the general rule is to hyphenate the word where the syllables join. If a word has no syllables do not hyphenate just carry the entire word to the next line. (e.g.: Chari-table Foundation)
You would hyphenate "thank you" when it is used as an adjective before a noun, such as in "a thank-you card."
No, you do not hyphenate a double consonant word when it is used in a compound word or as part of a phrase. The double consonant remains intact.
You don't
yes
You can hyphenate the word improvement like this: im-prove-ment.