Copyright law doesn't protect titles, names, short phrases, or slogans, and games must be new, useful and not obvious to get patent protection, so your intended use may be allowable. You should seek qualified counsel to advise you before you spend the time and money developing anything that may infringe the rights of others.
Disclaimer: Please note that this answer does not constitute legal advice, and should not be relied on, since each state has different laws, each situation is fact specific, and it is impossible to evaluate a legal problem without a comprehensive consultation and review of all the facts and documents at issue. This answer does not create an attorney-client relationship.
As Ms. Koslyn notes, short phrases are not protected under copyright law. This means that because copyright does not attach to short phrases it is not a copyright violation if someone other than the author reproduces and/or publishes the phrase. So, in your situation, the various authors of the phrases cannot use copyright law to stop you from reproducing and printing the phrases you want to use with your board game.
Short phrases can be protected under trademark law, however. Even so, it's difficult to see how (for example) printing cards that contain a trademarked short phrase would be an infringement of the mark. It WOULD likely be an infringement to use a trademarked short phrase as the name of your board game and there may be other ways that you could infringe as well -- depending on precisely how you use the phrases.
As noted, you need an intellectual property attorney's advice before you use in commerce any material that was -- or will be -- created by someone else.
Depends on the length of the tongue twister, the originality, and whether it is an old expression or a newly created one.
Each case is specific and must be evaluated on its merits. It the saying is old, then it is fair use. Check that it is not use as a trademark.
DISCLAIMER: This answer does not constitute legal advice, and should not be relied on. Each question is fact specific and requires a comprehensive evaluation and consideration of all the facts and documents at issue. This answer does not create an attorney-client relationship.