Is giving a girl a rose on our first date a good idea, even though our date is on Valentines day?

Of course you should give her a rose and would be bad if you wouldn't do that, because it is valentine's day. Each girl likes attention, small gifts, flowers... Just you overdone with choosing the color, when to give it how to do that. It doesn't matter ;)

I think it's sweet. I wouldn't worry about matching a dress. Just give her one red rose.

its totally awesome....and if she thinks ur a creep or acts creepish toward u then just know that even in fact you did purchase the date for a good cause, she did put herself UP for the challenge. the flower and all sounds very sweet, just make sure to relax and try not to say the first thing that comes to mind which could creep her out. and if you really like her after getting to know her it could bet the start of a nice friendship. Honestly the first 10 min lets u know if ur happily comfortable around someone.

It depends how she perceives it. Getting a rose or other flower the same colour as her dress would show that you're thoughtful, but it could also show you're confident (perhaps too). I think it mainly depends on how you deliver it. Whatever you do, just don't pass it over the table. It can make you look smug and arrogant.
My personal opinion on when to give it would be just before you say good night to her (whether it be when she gets in a taxi or if you drop her off at her house), tell her you have something for her. Pull it, but not too quick. If you bring it out at a moderate speed then it'll show you're not full of yourself and confident in everything you do. The fact that you can present a rose will make you look confident, but like I said, watch how you present it as it will highly influence her view of you.

I think it really depends on the type of girl she is and also how do you present the rose/s to her. Everyone perceive things differently, and whether you give her a big bunch of roses or just a few roses to spice up the evening could make a huge different too.

Personally I don't think any girls would reject flowers, it's just what you do with it and your intention really. So practice your speech after you give her the flowers. Work on your charisma instead of worrying more on the actual roses, afterall you are not going to just give her the rose and that's it!

It's too cliché if you ask me...