One thing - when I was young and single, I decided it was only fair that I ask men out as well as having them ask me out. When I tried this however, it NEVER worked. NOT ONE TIME. ONLY if a man had asked me out at least once could I reciprocate and have things go smoothly.
My take-away? While men may complain about needing to take the initiative, when a woman does it, it doesn't work.
More recently, after my divorce, I tried this again (asking men out) with the same results.
Once a man had asked me out a couple of times, after that if I started initiating, it was fine. Also, re. paying - some men prefer to pay for everything. Some want to split the check 50/50 even if they asked you out and ordered 3 drinks to your 1.
The reality is that everyone is unique.