K.
Remember... 40 weeks is your ESTIMATED DUE DATE. This means that 50% of moms give birth before the date and 50% give birth after the date. It is not a deadline. Your baby will come out when it's ready.
I am currently 41 weeks 1 day pregnant and am patiently waiting my baby's birth. Once you get past 41 weeks they do recommend that you have a non-stress test (basically put you on an EFM for 20 minutes to make sure the baby looks fine) and they check fluid levels via ultrasound. If your baby looks good there's no reason not to let it cook a bit longer, and these things will give you a good indication of whether the placenta is aging or continuing to do it's job.
Statistically, the risk of stillborn is about 3/1000 at 40 weeks, about 4/1000 at 41 weeks, and about 5-6/1000 at 42 weeks. So yes, the risk goes up but not enough to worry about.
Ok so induction... Personally, I think if you're anxious and you're already somewhat dilated you can opt for something like a membrane sweep. This may encourage your body to go into labor a little earlier than it might otherwise, but it will not work unless you were close anyway.
I would NOT do anything else RE induction until you get to at least 41 weeks (unless there's a concern with the baby). Inductions frequently initiate a "cascade of interventions" and that can end in a birth experience that you don't really want. Being induced isn't fun I hear. Artificially-induced contractions can be stronger, more frequent, and more painful than naturally occurring ones. Unless there's a medical reason to induce I think it's best to let the baby come out on its own.