@Unfashionable
Absolutely not. It would reassure me that he sees women as equals.
Not necessarily.
In my experience - and that of many women I know - a man who wants a woman to pay her way, is NOT wanting her to pay for herself because he thinks women are 'equal' and he supports 'equal opportunities for women.'
You will find men who are all 'women fought for equality so they can have it and pay their own way,' are not so keen on equal opportunities for women when it comes to pulling their weight around the house, and sharing the domestic, and childcare duties.
I am happy to pay my way in life, and have never expected to be 'kept by a man,' (urgh I HATE that misogynistic phrase,) or treated like a precious princess. But, like many women on here, if a man asked me on a date, and at the end of it, he started reckoning up what I had had, and telling me the bill is £33.00 and what I had comes to £14.23, it would be the first and last date. This behaviour = tightwad IMO.
I would offer to pay my share, but would still be massively put off if he agreed, and said 'yeah let's work out how much you have to pay.' If he DID pay, and wanted to pay, this would be a good sign (IMO,) that a man is chivalrous and kind generous, and will look after you.
As I said, I don't need looking after, and have always brought income into our family, but it's nice to know he would look after you if you needed him to. If he pays, it shows respect towards the woman HE has asked out. Asking a woman out, and then asking her to pay for what she consumed? Urgh! How off-putting! 
If the man DID pay, and was happy to pay, then on the second date, I would offer to take him somewhere, and pay for it...
And no, I am NOT explaining why I think how I do. I don't owe anyone an explanation. So the ones demanding to know why people have the opinion that they want the man to offer to pay on a first date; don't waste your breath. The OP asked. People have answered . No-one has to explain their opinions.