Firstly, you should talk to all your players about the issue outside of a session. You can get some personal impressions first with one-on-one conversations, but ultimately the entire group should sit down to discuss the problems. Make sure the discussion is democratic in nature, though. JohnP points this out in a related question: "The group setting can be dangerous, as it can turn into people digging up old grievances or ganging up on a particular person."
During the discussion you should ensure that everyone is on the same page about how they want the game to be played. The same page tool is a useful set of questions that will drive the discussion in the right direction. Just make sure everyone is fully involved in the discussion and are voicing their opinions clearly (no passive-aggressive bs). If your players are open to compromises a consensus should be reached.
Secondly, make sure your friend is not suffering from My Guy syndrome. If the description matches (and it sounds like it does), show her the link privately and let her think about it. It should help her be more aware of her role as a player and hopefully remedy some of the issues.
You also mentioned your player gets upset as a result of your decisions. This related question contains a lot of suggestions for how to handle players that take things personally, ranging from studying your own approach to kicking out the problematic player.
In the end, though, you are the GM. You are the writer of half of the story, the referee on all mechanics, and the leader in the quest for fun. The way you drive your campaign is part of your style and your players should respect it. They need to be aware that your goal is always to increase the enjoyment they collectively get out of your game. A certain level of trust and respect is absolutely necessary. If this is impossible to obtain in your group, then the group as it is cannot function. Kicking out players or stepping down as GM would be the next steps to try.
Because the GM is so special, though, new players often fail to grasp just how complex the GM role can be, and can at times see him as an enemy and spoilsport. A neat "trick" you can use to show your players what being a GM is really about (that also gives you some rest from the responsibility of being a GM) is to have another person in the group be the GM for a few sessions. It doesn't have to (and most often shouldn't) be the same campaign you're running. Instead, it can be a few sessions of an off-shoot campaign. It's fun for the players because they get the chance to try out new (and often times silly) character builds, and the new GM will discover what it feels to have all this responsibility. Once everyone has GM-ed a couple of sessions, you will all have an idea of who's best at it and will work towards keeping that person as GM in the future. There's a chance it might not be you, but in the end it should result in a better experience overall.
Keep in mind that not everyone is fit for GM-ing, or willing to try at all. Don't force players to GM, and if they decide to try, encourage them to design very short adventures (no more than three sessions). They can always expand on them later if they like it, or end them early if they don't. The player that questions your decisions often probably thinks they can do a better job, so they're likely to accept your offer to prove themselves.
A Constitution saving throw is fine
Constitution saving throws are already used when checking if a spellcaster's concentration breaks when hit by an attack, so they can be considered an established option for checking if one's mind stays focused when in pain.
Alternatively, you can use the variant rules on Player's Handbook page 175 and make it a Constitution (Deception) check, which works like a normal deception check except the attribute modifier added is the one of Constitution, not of Charisma.
Best Answer
There is no set skill for seduction. It's going to fall back to a discussion between you and your DM on which ones you feel appropriate, with the DM having the final say.
Personally, I don't think it's actually tied to any one skill, it's going to be the overall nature of what you're trying to do.
Trying to seduce someone you have zero actual interest in, as a strategy point, a way to achieve a completely separate goal? Sounds like you don't actually have a (direct) interest in bedding him/her, so Deception.
Trying to seduce someone because they're beautiful and you just want in his/her pants on general principles? Persuasion
Oh, wait, you're using music, song, poetry, and other Performance type things as the core of your seductive effort?
And while I certainly hope you're never this gauche, this cruel, trying to convince someone that his/her only hope of survival is to partner with you sounds very much like an Intimidation effort to me.
It's also worth noting that a clever player or DM could come up with even more interesting, less obvious combinations. An NPC who's attracted to muscle men may take an Athletics check; you may peel a lime in a single, long strip with Sleight of hand to suggest your abilities in the bedroom; a creative player or DM could probalby find a way to be suggestive with a LOT of skill checks -- or even a flat out ability check.