My third daughter was born at 37 weeks and was fit as a fiddle. My oldest was born at 42, and needed some intensive immediate care for meconium aspiration. Heck, I was born at 32/33 weeks and had only mild breathing difficulties.
It all depends on the baby. 37 weeks is considered full term, not a preemie, so it's as safe as can be expected. Even a baby at 40 weeks can having breathing difficulties, the risk goes up the farther from term (37 weeks) you get, as the lungs haven't matured.
Princess...that's [I should say sounds like] sciatica, as girlfriend noted. Baby is pressing on your lower spine. I used to get awful sciatica through my hamstrings.
Unfortunately for you, it can be a signal that you'll experience "back labor" (when the baby's engaged head causes increased pressure on the spine during a contraction). If you had any notions of going au natural for your delivery, you should start getting them out of your mind. If you have back labor, you'll be screaming for an epidural by the time your contractions are 4 minutes apart.