I am of the firm believe that when a participant voluntarily demonstrates a lack of respect towards an umpire, it's usually time to toss them.
In your first scenario, the player did demonstrate a lack of respect towards your role in the game. However, we're talking teenagers here, and anyone who has ever been around teenagers will tell you that they're, well, not exactly well-restrained. Her comment towards you about ejecting her was a little on the impulsive side, and, in a way, possibly involuntary. I think that you probably handled her comment about ejecting her in a fashion that's best suited towards that level. I think you should have had a chat with her coach that goes something like this...
(In a VERY calm and collected fashion) "Coach, I realize your player is upset, but she can't make comments like that again. She needs to cool down, or she will not be participating in this game any longer."
(Edited out the part where I addressed the call, as it only baits the coach to argue her side. Frankly, the call itself is irrelevant.)
I do think, however, that you let her get the better of you by baiting you into arguing with her. I would've done this...
F5: What? I was on the bag!
Me: No, you didn't have contact with the base.
F5: Yes I did. I was touching the base.
Me: Player, that's enough.
If she objects further, call time, pull the coach aside, and have the above talk with him/her.
As for the second scenario, the coach pushed the line pretty hard. It's a bit tough for me to judge the coach's tone, but I'm about 60/40 towards tossing him. A comment like that is a direct insult towards an umpire, and a complete demonstration of a lack of respect towards your role in the game. Whether or not he's correct in his understanding of the rule is irrelevant. What's relevant is whether or not he conducts himself appropriately towards you, and his comment was definitely directed towards you.
Okay, never mind. I just talked myself into it. I'm now 100% for tossing that coach!
I agree with Mike in that keeping a streak alive does yourself no favors. The perception that's given is that you can be a pincushion, and when you need to stand your ground and make that tough call, it will be ten times harder to do so.
Take not my comments personally, as I've also been in the same position before. In the beginning, I tossed too many. In the middle, I probably tossed too few by having the same goal: trying to keep someone in a game. Now, I think I've found a happy medium of keeping only the deserving participants in the game.