Wow, what is wrong with asking a question about dating on a parenting site? Are single parents not allowed to date or rebuild their lives? There is nothing "odd" about that. It's a relationship question, whether it pertains to a boyfriend, husband, sister, or in-law...Sorry, that comment just irked me. Anyhow, on to your questions:
Typically, the man should pay on a date. Call it old school, chivalry, whatever, but in these times when women are also working, you may want to offer to split the bill, or pull out your card. Chances are, 98% of the time the man will tell you to put it away and will pay the bill himself. This shows you're considerate and fair, rather than high maintenance and expecting that it's his "duty" to do so under every circumstance. If you're buying something for yourself though (a necklace at the museum gift shop, an ice cream cone even though he didn't eat any of it because he doesn't like ice cream), then be prepared to pay, as most men won't do so unless it's something you shared, like a meal, or a movie outing. Most guys have told me that if they're engaged or married, then at that time they have a "I pay this time, you pay next time" rule, but not during courtship when they are trying to make a good impression on the lady.
As to physical intimacy, this is something very personal and no one can advise you on that, you just have to do what you think is right and appropriate for YOU. Yes, there are many people who had what they thought was a one night stand and ended up engaged and married for over 10 years, but there are also those who had sex on the first date and the guy stopped calling. The same applies to people who have dated a month or 2 and then had sex, because if the chemistry is not there or there is a lot of friction, regardless of the timing of intimacy, the relationship WILL end. Also, poor sexual chemistry can ruin a relationship as well. If the other person is a cold fish in bed or cannot perform, this can cause strain on a relationship, or if one of you is sexually active while the other is celibate or not very interested in sex, the same can happen. You have to decide how important sex, as well as good sex, is to you. If a relationship is meant to be, I don't think a particular timing decision regarding intimacy will change that.
I try not to use internet dating sites due to bad experiences, plus I know plenty of men who scour those sites for fresh meat. I have signed up to meetup.com and have found groups of people that do things I like, and while doing those things, I have met some nice people. You can definitely build up a relationship from this point if you have good chemistry and you both have similar interests. What better way to find someone who likes what you like than meeting them in that setting? Most of these people are looking to make friends who like what they like, they are not really there to pick up their next booty call or get married, like on a dating site, so things are less forced, more natural, there is no pressure, and things can progress at a leisurely rate because it's not a dating site where you're one on one, it's a group of people and if you like a few of the guys and gals, you can then go have a drink and get to know each other more and get phone numbers to stay in touch so the decision to go out with this small group or one of the people of the small group is entirely up to you.
Good luck whatever you decide to do, and I do advise you wait to introduce any potential boyfriend to your kids. I am not sure what the appropriate timing is on that, but personally, I'd wait 6 months or even longer to do so, as people can enter and exit your life constantly and you don't want to send the wrong message to your kids, or give them the impression you are unstable. I know some women who introduce men to kids within a week or 2 of dating and I think it's a bad move, considering that the guy isn't even a boyfriend at that point and by the end of the year, the kids have met 8 or 9 men.