I searched here and found some info but I'm still a little confused about the notifications. Here's my understanding so far:
1. Board notifications - If you check this option, you'll get a notification when a new thread is posted. In order for this to work though, you need to have the entire board marked as read.
2. Thread notifications - If you check this option, you'll receive a notification when a new reply is posted.

So my questions are:
1. Does the board notification send a message for EVERY new thread or just the first new thread?
2. Does the thread notification send a message for EVERY new response or just the first response?

I tried testing the thread notification on my own site and it would send me a notification for the first response but then deleted the thread notification from my profile. Is that normal or am I doing something wrong? Do I really need to select "email and watch" every time I read the thread?

Just when a new thread is posted since you last read the entire board (or read the thread, if it's a thread).

Nothing should be deleted from your profile when you reply to a thread.  Did you merge or split the thread?

I hope this answers your question.  

We're still having issues with the thread notification sending messages on a consistent basis. Sometimes members receive them, sometimes they don't. Is it supposed to notify for every new post on a thread?

The thread notification was definitely deleted out of my profile when I read the thread.