1. Yes. When you publish a course with multiple quizzes for an LMS and indicate that the user must pass a quiz, the program will send the LMS info from whichever quiz is completed first. Since your learners would only take one quiz, its results would be submitted to the LMS.
2. You can't rebuild the Player Menu based on which path the learner chooses. (At least, that can't be done with just Storyline. It might be possible with special coding, e.g., JavaScript.)
FYI: This post is about creating different paths. It includes a comment that has a demo file for creating a custom menu for each path. TIP: Let the user choose their path - Articulate Storyline Discussions - E-Learning Heroes