Your problem is not that your husband has a woman friend. My husband has women friends - some are married and some are not. I have male friends - some are married and some are not. He has lunch with his female friends, I have lunch and even have traveled on business with my male friends. Their status is irrelevant because people can cheat or be faithful regardless of marital status. But the difference in our case is that we have a strong marriage and a trusting relationship. We tell each other who we talked to and what they said. Neither of us would dream of telling the other not to talk to someone.
Another person NEVER breaks up a strong marriage. It's always a fault in the marriage if things fall apart. The 3rd party can be the temptation that causes one partner to cheat, but it's the partner who does it, not the other woman. So your jealousy is way off on that. I also think your comments about her appearance are irrelevant and petty. In that sense, you would help yourself if you were more secure.
You say you are secure but you don't trust other women - that's a contradiction. You don't trust your marriage and you are constantly worrying about it. You've been taught to keep your man away from single women and your girlfriends? Someone who was very hurt has passed that on to you, and it's crippling you. It's not healthy, and it's not normal. What's more, you're going to pass this on to your children, teaching them to be suspicious and letting them know that most people are cheaters if they aren't policed and watched all the time? That's not going to work out for your kids in the long run.
In that sense, I can't blame you. Your husband is giving you lots of reasons to not trust HIM. Your husband is out of line because he doesn't seem to tell you when he speaks to her. When people keep secrets, it's the sign of a breakdown in the relationship.
And I'm really alarmed that he asked you to "do him a favor" by watching the 2 teens for 2 weeks. Why is this a favor to HIM??? That's a huge red flag. He had absolutely no business even discussing this with his friend - he should have said no, but if that didn't happen, he should have gotten all the details and then said he would discuss it with his wife since 90% of the responsibility would fall to you. That didn't happen - and that's a huge problem. I agree that no one should be leaving the country for 2 weeks without having made 100% arrangements for the children, and that would involve sitting down with you, not just your husband.
But the biggest problem is that you and your husband don't communicate. You can't speak without someone getting angry. You don't trust him, and he doesn't consider you enough of a friend to even level with you, let alone consider your feelings.
You need marriage counseling, but not because of this woman. It's because you and your husband don't respect each other.