Just click on them to view and they will be removed or at the top of the list theres a tab on the right hand side that says mark all messages as read, that will clear them all.
You can't, if you participate in a thread then it will show in list of "new replies to your posts", you can also put yourself on alert for a thread by pushing the notify button right next to reply; this will make RomUlation email you when there's a new post.
