Hey dude!
I work for a large retail bank who on the surface would certainly *seem* to be a modern, egalitarian sort of place to work. The regional manager who is responsible for most of the hiring told a colleague over lunch that he'd not give a job to a girl he wouldn't want to screw. On the other hand, I've seen females flash a bit of cleavage at a sucker manager/customer or use a bit of strategic sexually suggestive body language to get some kind of desired result.
I don't think the workplace has changed as much as you imply and neither has the dating scene. It's just got a bit more sophisticated, and maybe a bit more complicated and ambiguous for men who are now expected to simultaneously exhibit positive female gender traits (sensitivity, use of moisturiser) whilst retaining the best male ones (physical protector, provider), and the flexibility to know when to switch between modes. Ultimately I think think 'pure' equality of the sexes is an idealistic concept.This is not right or wrong, but an expression of our biological imperatives. Cultural norms will dissolve, reinvent themselves, mingle and transcend, but always as an expression of our different biological natures interacting with a changing environment.
Regards to the girl at the store: Do you think she'd be comfortable being asked on a date by a customer, because currently that's your relationship with her. Maybe, maybe not. You don't want to put her in a position where she might think you're cute but she says no because she doesn't feel safe going on a date with somebody she doesn't know well. She might panic and say no, simply out of fear of the unknown. If on the other hand you have good rapport and she is showing positive signs, and seems game, just go for it. It really depends on the girl and how she perceives you. If its a bit ambiguous, maybe dilute the risk she is taking with you by finding out where she socialises with her friends and going with a group yourself. It won't put either of you under the spotlight so much!