Unfortunately rejection does happen. I know it sucks, especially for someone with anxiety because it's a big blow to the ego. But isn't it better to get that over with right away, versus waiting for weeks to find out that she's not interested? It sounds like you were able to move past it and have a good friendship with the girl who turned you down.
Personally, I don't recommend dating coworkers, since it can become awkward if things don't work out. Plus you spend a lot of time with them at work, so then when you add personal time, you're not getting a lot of time to yourself. For me, that can be a recipe for disaster. Now that being said, it is perfectly normal to develop crushes on people that you work with, because you do spend so much time with them. I think you've hit the nail on the head there. You combine spending a lot of time with a girl at work, plus recovering from a previous rejection and looking for something new, and that opens you up to develop feelings for another co-worker. I'm sure this girl you like now is a great girl, don't get me wrong, but I think (coming from an unbiased opinion and someone who does not know you personally) that it's more the other reasons that you're finding yourself attracted to her.
Given the fact that you have asked out a couple of girls at work already, it may be a little awkward to start dating another coworker. So far as your worry for whether or not her friends would like you, well that's a hurdle we all have to face when beginning a new relationship. Meeting the friends and family can make us all nervous, anxiety or no, since we all want to make a good impression. The best thing you can do is to be yourself and hope for the best instead of the worst, hope that her friends see the good in you that brought on her interest in you.
No judgement here at all, I have generally dated guys who were older than me in the past. But I didn't start dating guys with an age difference until I was 18. I personally see no issue with age difference in dating, so long as it's reasonable. But keep in mind that at your age, four years can make a big difference. I'm assuming that, because of her age, this girl is still in high school, where you're likely finished. This already puts you into different worlds. I'm not sure where you're located, but in Canada you would be legal drinking age and she would not. In the US, a year from now you will be legal drinking age and she still will not. This means that, for a few years, there are things that you may want to go out and do that she cannot.