I currently work for a law firm that’s unaware I’m applying to law school this cycle (I’m scared I will lose my job if they know I’m leaving next fall). They are listed on my resume and my personal statement but my boss is not a recommender. Do law schools typically contact employers about the applicant?

I took the August LSAT a few days ago and am planning on retaking it in October just to be safe. Should I apply as soon as possible, even if I don't know all of my scores yet, or submit at the end of October once I know both scores?