In this context, I think it is some kind of biological feature that locks the jaw bone in place. "Latch" here means "to lock in place", like a latch on a door （门闩）.

Ça dépend de la découpage en syllabes du mot. On trouve /ø/ dans les syllabes ouvertes (où la voyelle est à la fin de la syllabe) et /œ

"Recently I completed an orthodontic treatment where I was going to the odontologist monthly. I am currently using a retainer. In general I have good oral health."

You can be a mentor to someone or be a mentor for someone. Both are correct and mean exactly the same thing. You can mentor him on selling products or you can mentor him about selling products, again both mean exactly the same thing.

For the second sentence, both are correct, but they mean slightly different things. "She told me we had stayed in the same hotel the previous summer." "The previous summer" means the summer before she told you this. If the conversation happened recently, it could be last summer, but if the conversation happened in 2011, then "the previous summer" would mean the summer of 2010. "She told me we had stayed in the same hotel last summer." "Last summer" always means the most recent summer before now. If you say "last summer" today, you are definitely talking about the summer of 2015.

Mandarin can be written in either simplified or traditional characters. It is not a writing system, it is a way of pronouncing the characters that is used both in Mainland China and in Taiwan. Mandarin is called a "dialect" of Chinese, as opposed to Cantonese, Shanghainese, Taiwanese, etc., though these can be thought of as separate spoken languages (they use the same written characters for the most part). They are not "dialects" the way other languages have dialects; speakers of different Chinese "dialects" cannot understand each other, so they usually use Mandarin to communicate with speakers of other "dialects".

Both are correct, you can say it either way.

"Labor" is a more abstract concept than "laborers". "Laborers" usually refers to the people doing the labor, while "labor" refers to the laborers' ability to work. When "labor" refers to people, it is usually in a political context, when workers' unions are organizing for a political cause. For instance: "The minimum wage increase is supported by labor, but opposed by Wall Street."