As In–Lecture Quizzes (ILQs) are just a special kind of quiz in the Quiz Generator, faculty and TAs have access to all of the normal views of student activity and grades that they have for non-ILQs. You can see the score achieved by a single student, and can get aggregate data about how all students performed on an ILQ from the Statistics page in the Quiz Generator.
As ILQs are designed for formative assessment, students can take the ILQ as many times as they want without a time limit, and students can take the ILQ after the "due date" for the quiz. As such, there are no special permissions for ILQs and no Special Permissions view is shown in the Quiz Generator for an ILQ.
You can set up an ILQ to link to the Gradebook so that you can track student completion of the ILQ. However, ILQs are only reported as being "complete" or "incomplete" to the Gradebook. That's the same as saying students either receive full credit or no credit for the ILQ in the Gradebook. There is no partial credit (e.g.; "The student received a score of 75% on the ILQ") reported to the Gradebook.
Students can answer questions as they view the lecture, or wait until a future point to answer questions. Students can watch the lecture as many times as they want and decide when they want to answer the questions to the ILQ. However, a lecture section with an ILQ will not be considered "complete" for tracking purposes until the student watches 80% of the lecture (which has been the rule in CoursePlus for years) and completes the ILQ. If a student just watches the lecture but never completes the ILQ, the lecture section will not be considered "complete" for tracking purposes. If a student skips around in the lecture to complete the ILQ but never watches 80% of the lecture, the lecture section will not be considered "complete" for tracking purposes.
The report that shows which students have accessed which lecture sections also includes information about when a student completed an ILQ (if an ILQ is associated with that lecture section). The lecture viewing report available to students shows them which ILQs are complete and which are still outstanding.