She is my follower. If I tell her that I want to part ways, where will she go?

Probably back to Dragonsreach.

I believe she won't be your Follower but will still be your Housecarl and will go back to your Whiterun house.

If you own a house in Whiterun she'll return there. If you don't she'll go back to Dragonsreach.

Your house in Dragonsreach/winterhun.

I had her waiting in my Whiterun house. While adventuring I got a message that my follower was tired of waiting and left. Will she still be there? I was too busy to check right then and now I'm at work.

If the follower is not a housecarl of yours (for example faendal) they will go back to the place you found them (faendal goes back to his home in riverwood)

If you leave a follower waiting anywhere you'll eventually get that message. Even though you left her waiting in your house she hadn't officially parted company with you - parting company only happens when you tell the follower to or when they've been left waiting somewhere too long.

I never dismissed her, but during the early companion quest I was forced to get another follower. So she said she'd be waiting for me and headed to Breezehome. She was there when I finished the companion quest.