If she initiates the friendship after the rejection, then you can consider if you can handle being simple friends with someone you have feelings for. If you have to initiate the friendship, then she doesn't really want to be friends. Just move on. Don't be a doormat.
Yes get continued to talk with her
Sure you can talk to her, but also it is up to her if she wants to talk to you. You can still be friends after what happened, but friendship should also be two way and she should communicate back. Youll never know she might develop feelings for you. Good luck.