I've read your post a few times and each time I come to the same possible conclusions:
1) She is flirting with you and very interested in you. What better way to connect than to link into the one very emotional vulnerability in your life?
2) Management is not being friendly to you and this is a probe to see what can be done. To eliminate this possiblity, skip any thoughts of coming out at work.
3) She is just a busybody interested in pulling people's strings and seeing what happens to them. This is not fun.
4) She is just a harmless gossip and would love to use you for gossip fodder.
So, 2 is soluble by not coming out at work. 1), 3), and 4) are soluble by taking the lady to lunch and sounding her out without committing to anything yourself.
I would move carefully, but in a direction to see what this lady's intentions are, truly. Then you can make your own decisions.
my
tina