16 weeks pregnant and having sharp shooting vagina pains in middle of the night almost every night. baby growing? nerve problem?

Not likely : At this point the baby is fairly small and uterus a little above the pubic bone so not likely to be due to size or nerve compression, which will often give pain down into the leg. The ligaments that support the uterus and other pelvic structures do become more lax and even the pubic bone itself will loosen and this can cause pain. Speak with your OB to confirm this and rule out other causes.