你同意教我开汽车吗？

应该是：Do you agree to teach me how to drive?

Drive本身就有“开车”的意思，不需要表达为drive a car。例如，司机就叫 driver，不需要叫car driver.

Do you agree to teach me driving cars?

Would you mind if you help me learn to drive?

你同意教我开汽车吗？
Would you promise to teach me driving cars?

agree + with sb. 同意某人
agree + to sth. 同意某事，某人的（看法，想法）

Do you agree with me to teach me driving cars?

Do you agree to teach me to drive cars?

Could you please teach me how to drive a car?