If she is dilated and baby is down low enough then yes, you can feel baby's head.
Also it can be up to 2 weeks past her due date before little one comes. Due dates are just estimates.
If the babys head was that low in her vagina then I would suspect that she would be in agonizing pain and sex to induce labor would not be necessary because she would be ready to push the baby out. If she is not screaming, moaning or complaining of pain then it is probably something else.
You're not touching the baby's head (unless she's dilated to 10 cm and is feeling the urge to push). You get deeper penetration in that position, try a different one. It's normal to "bump" into things during pregnancy. No worries. It doesn't mean anything other than your wife is pregnant and due any day. Oh... and, no, you won't hurt the baby and you won't be pushing on its soft spot. Unless your wife's doctor said no sex, sex is fine - just be sure your wife is comfortable. Have fun trying to induce labor - yours is definitely the most fun way to try!
Her vagina could be swollen but since the baby is behind the cervix I doubt that is what you are feeling.