Yep. And they are taking a lot longer to get these days since you will need them to leave the country period in the new year. so go get the job done STAT
Yes you do and you will receive one in about a month. But you can pay an additional fee to received them within two weeks. Make sure you have all of your info: 2 id pics, birth certificate, application, and application fee (approx. $110)
Yes, better apply for it and you will get one. Maybe, you're asking for the VISA. I think, no need for visa it yo will just stay here for only a week. But to make sure, once you apply for your passport, inquire about the visa, too.
Happy trip :)