#1
Is there a way to know when someone has replied to your thread,quoted you?
#2
go to my profile in the top right hand corner, then go down to forum post total, click on that, and then a list of all the threads you have posted in, and if there's a yellow envelope there is a new post, as for if someone has quoted you you just have to go through every post
#3
If someone has quoted you, click the search button next to the new posts button. Search for your username, and select "show results as posts"

If someone has replied to your thread, you can do what britishsligean said above, or, on every thread below the reply box, there is a link you can click to subscribe to the thread, so that you get email updates. You can edit your subscriptions through the control panel, to select instant, daily or weekly emails, and you can also set it to autosubscribe as soon as you post in a thread.