They are held about a half dozen times throughout the year. The admissions site usually has the dates for the year. Be surprised if they listed them for next year yet. There is not a certain % required. There are juniors currently attending them who don't have an open application. Their data has been pulled from STEM and NASS applications. Talk with your BGO and they should be able to help guide you.
DS is currently CPR. Last year as a junior he applied to NASS when it first opened in mid January, and got a slot. About a month later he received an email from Admissions inviting him to a CVW, which occurred in April. Outstanding time at the CVW. Then NASS in June, even better. He has previously applied for USNA Summer STEM, and was rejected 2x 9th and 10th grades. We we told not to be upset about the STEM rejections, that it is essentially a minority outreach program (DS not a minority applicant ). Good luck.