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Slow down. Take a deep breath. Relax.
Yes, call her. When you call her, ask her "Who is this?" and when she replies say "Oh! Hey! You're that girl from the party, right? I found this number in my pocket, but it didn't have a name attached to it."
Yes, you should ask her out, but not directly. Tell her you're going out for coffee, and as an afterthought, invite her along. Or, my preferred method, simply order her to come for coffee with you.
Method1: I'm going out for coffee in an hour. You should join me. I'll bring my slinky.
Method2: You're joining me for coffee. I'll bring my slinky.
If she says no, don't worry about it. Call again a few days later and repeat as neccessary. As for where to go, go for coffee. Or go window shopping. Or go to a park.
No, you don't pay for anything. ANYTHING. Until you've seen her naked, you owe her nothing. Just start up a conversation and expect her to pay for herself if you're doing something money-related.
Nice.
I'll add that you should mention your name in the first 10 seconds, because I guarantee she forgot yours too. Helps build rapport.
"Oh hey! It's Joe! You're that girl from the party right?"