My answer is no to the flowers, but opening car doors is ok.
Buying flowers is shows you are giving too much effort on a first date and she doesn't even know you. You are acting too desperate.
Opening car doors is a nice gesture that can be done if you want to reward her for good time, but don't go out of your way to do it because it is just too polite given this new era of relationships.
The bottom line is, on first date, you are there to have a good time as give her a good time. Appearing to make too much of an effort on a girl that you don't even know makes you look desperate. Best to make everything look natural and easy. Only start to show effort, when she shows she likes you first!
Don't make it look like you are making an effort, but do have fun and do make sure she has fun as well.