I'll try to help.
what do I charge her?
$100. Seems to be the magical number for some reason with small jobs.
If your not comfortable with that. Do it for half.
Should I give myself an hourly wage or should I think of an overall price for the product?
Price for the product. Hourly wage comes with huge projects with multiple assets for a project.
Do I ask for payment up front or do I wait until I've shown her the drawing?
Show her your drawing and then hold it ransom.
How do I ask for payment?
If you haven't asked yet....Its like any other negotiation. Just straight up say it. "Hi So'nSo', I started working with your graphic for the blog, I worked out the price for my time and its going to be $75." Period. See what she says.
When I'm communicating with her, do I try to sound professional or do I let myself be relaxed since we know each other?
If you know each other relax. I've done work for friends before, we talk and relax, as soon as we talk about money, we do it for 2 or 5 minutes, and its over.
Since I'm creating a drawing that she'll use on one of her projects, am I supposed to have a copyright or something?
You get that automatically in America. Sign it. You own it. You can tell her that she is buying the internet usage rights for the image. You dont have to tell her that you retain the right to use the image for anything else that you want to use it for. Sell it to other people if you want to be a hustler..... Anything she wants to print or make with your image on it though she pays again. See that's the difference that a lot of young artists don't know about. What companies are actually paying for is the "one time printing rights." or 'Exclusive Rights' Meaning if they print it again for a second run, or another product, they have to pay you AGAIN. Or they own it and can print it on whatever with exclusive rights privelage, but guess what, they have to pay more for that.
Do I need a contract?
Do you think you need one? If not don't worry about it. You know her, she knows you, she'll throw you the $100, and be done with it.
What do I do if I show her the drawing and she doesn't like it?
Dont give it to her.
What if she changes her mind?
Your still friends right?
How do I make sure the drawing looks good enough to be seen on her website?
Look at other website formats that use the same template, ask her what she's using to make her site. Blogspot? Wix? Etc, look up other sites similar to see what she needs.
How do I format it so that it looks good for her?
Scan it at high resolution and color correct it on a computer.
Should I be worried that I messaged her that I would be interested and she hasn't gotten back to me in two days?
Nope, move on or work on it. And then keep drawing those faces.
Cheers.