If you ask 90% of modern women they will tell you chivalry is not dead it's just awkward as f#@k. Whilst saying that, 90% of women also say that having the door opened for them on a first date is a very welcome act and will have you moving in the right direction of not being perceived as a total Douche Pirate. Follow these acts of modern chivalry and have her saying "Well, I never!" in no time.
When was the last the you turned down a free dinner? The answer is never. By offering to pay you show that you appreciate her going out with your hairy ass.
Open The Door
Unless she has PTSD from being attacked by a doorman, opening the door for your lady is a definite good move.