A gap probably wouldn't just help you get a better LSAT score and get you into a better school, it will give you an additional year of maturity and experience that may make you do better in law school which will help you in the job market after.
3...i just checked u of chicago numbers you are well within the range and they are top 5. I would focus on writing stellar essays and securing solid LORs then apply. Congrats on 171.