Do I have to claim the $100 per week that my wife gets for taking our neighbor to work? If so, can I deduct mileage that she drive to get this money?

Yes and yes. This is considered to be self-employment income. She would need to enter the income on Schedule C. See the steps below. She can only take miles that have a business purpose. If any personal miles are included, she would have to subtract them out.

 No 1099-Misc HOW TO ENTER

1. First, sign in to My TurboTax
2. Next, click the orange Take me to my return button
3. On the white bar at the top, click Federal Taxes
4. Then, Wages & Income
5. In the Self-Employment section (you may need to click Show more), next to Income and Expenses, click Start (or Revisit)
6. When asked "Did you have any self-employment income or expenses?" select Yes
7. You'll see the screen titled "Tell us the type of self-employment work you do." Enter your description, then Continue
8. On the next screen "Let's get income for [description]" select the button Additional income. Includes 1099-K, check/cash payments and Continue 
9. The next screen will say "Tell us about additional income for your [description] work." List descriptions and amounts on this screen