Let's look at the rules under the assumption of good faith
Based on your opening paragraph, all of your players are comfortable with "sexytimes" role play in D&D. What you are asking for is something mechanical. Seduction and such isn't a thing D&D handles well mechanically.
Short Answer: the rules don't handle this well
Brief responses to your questions:
- What do the rules say about how to handle this situation?
There aren't any, for seduction, but you can shoehorn a few into this situation if your whole table is bound and determined to do so.
- What ability/attribute/whatever roll is required?
None is required, but if you all want to go down this route, then Charisma (Persuasion) is the closest ability check to Seduction.
Beware! There is a can of worms1 being opened here.
Beyond that, one can make the case that Seduction could just as easily be Intimidation, Deception, and / or Performance rather than Persuasion, or even a combination of some or all of them. After review of how Ability Checks are laid out in Chapter 7, a straight Charisma Check might be more appropriate (without modification by any skill/proficiency bonus) versus Wisdom might fit better, and it might not. Again, the mechanics for doing this in D&D don't fit very well.
- Do the rules say how long a seduction attempt should take and what
situational bonuses (or negatives) to apply to it?
No, the rules are silent on seduction, although some magical spells or creature abilities can have a similar effect. (See MM, p. 285, Succubus, Charm).
- To what point do the rules allow your DM to "take control" of your character during this seduction check?
They don't. As with item 3 above, the rules are silent about this. If magical effects are involved, and a save is failed, there is some "loss of control" by the PC until a successful save is rolled, or the magical effect ends. For an ability check, you can accurately say that once the dice are rolled, the DM narrates the result (Basic Rules, p. 3). That said, when it comes to interpersonal role playing, the less the DM has to say about this the better.
Why should the DM interfere with the role play between two player characters?
The DM makes decisions for all of the NPC's; the players make decisions for the PCs. If you all are fine with the DM taking charge of your characters, and reducing your agency, that's for the group at the table to decide. I recommend against that.
Why?
You are in the game to be a player, not a spectator. You, the players, make the choices and decisions for your characters. The DM has the rest of the campaign world to handle and make choices for.
Bottom Line: the rules don't handle this well
Seduction is not handled by D&D 5e rules; magical effects may have a similar result, but that still needs to be handled with care by the players at the table.
Amplification on the above: ability checks are not magical spells
- One of the players is playing an extremely attractive female character. She's a 1st level rogue with 18 CHA (maxed out, half-elf and the DM allowed it to get 1 bonus point in exchange for a pitiful 6 STR) who is well aware of her charming beauty and willing to use it without any kind of moral sense.
In the rules for this edition, Charisma is not a measure of physical attractiveness; it is a measure of the power of personality and personal magnetism.
- Aside: this edition of D&D chose not to fall into the trap that the 1e AD&D Unearthed Arcana chose to dive into, which was the addition of Comeliness, or physical beauty, as a character trait. There have been some lessons learned over the years. This is one of them.
If you all, as a table, want Charisma to point towards beauty feel free to do so. The game's rules don't cover that.
Charisma
- Measures: Confidence, eloquence, leadership Important for: Leaders and diplomatic characters (p. 8, Basic Rules)
- A character with high Charisma exudes confidence, which is usually mixed with a graceful or intimidating presence. A character with a low Charisma might come across as abrasive, inarticulate, or timid. (Basic Rules, p. 9)
- Charisma, measuring force of personality (Basic Rules, p. 57)
Under Ability Checks, we find
- Charisma Deception / Intimidation / Performance / Persuasion (Basic Rules, p. 58)
- Charisma measures your ability to interact effectively with others. It includes such factors as confidence and eloquence, and it can represent a charming or commanding personality. (Basic Rules, p. 62)
- Other Charisma Checks. The DM might call for a Charisma check when you try to accomplish tasks like the following:
• Find the best person to talk to for news, rumors, and gossip
• Blend into a crowd to get the sense of key topics of conversation (Basic Rules, p. 62)
Unless you all, as a table, agree that Seduction is a form of Persuasion (or Intimidation, Deception, etc) then there is no mechanic for Seduction. All of the above underlines the following point: D&D 5e isn't mechanically built to handle seduction, unless a magical effect (like the one a Succubus uses) is in play - and that gets a saving throw.
- Last time she tried to seduce my character.
This is a PvP interaction. Is your table already good with PvP? If yes, proceed. If no, all stop, simply say "No thanks" and play on.
But, if you all do agree that Seduction is a viable PvP event, and that you are interested in playing this out, then you can have a contested ability check. From "Contests" (Basic Rules, p. 58. Same words in the PHB).
... special form of ability check, called a contest. {emphasis
mine}
Both participants in a contest make ability checks
appropriate to their efforts. They apply all appropriate bonuses and
penalties, but instead of comparing the total to a DC, they compare
the totals of their two checks. The participant with the higher check
total wins the contest. That character or monster either succeeds at
the action or prevents the other one from succeeding. If the contest
results in a tie, the situation remains the same as it was before the
contest. Thus, one contestant might win the contest by default. If two
characters tie in a contest to snatch a ring off the floor, neither
character grabs it. In a contest between a monster trying to open a
door and an adventurer trying to keep the door closed, a tie means
that the door remains shut
- I'm playing a male (kind of, but this is for another post) paladin. He's not an "asexual" character, but he's very focused on his goals - he swore an oath of vengeance - and perceives such things like carnal pleasure as useless distractions.
The simple way to mechanically apply this is to (1) give this Paladin character advantage on the contest, based on his normal modus operandi, and (2) if I were the DM, also give the initiator disadvantage given how little interest your character has in such things.
But to be honest with you, that's roll playing, not role playing. Your character should be able to say "No thanks, I am not interested" and that's the end of it. It's the course of action I'd recommend; all of the rest of this answer is based on your stating in your question that "your entire table really wants to go there."
Experience
Over the years, I have found that at a certain point, person-to-person intimate or sexual role play needs to go off screen. The players who are not involved are reduced to spectating at best, or idly waiting for their turn to do something - and sometimes they'll feel uncomfortable with that situation arising during play.
When this little conflict escalated, the DM ruled that the seduction
attempt deserved a die roll. Nobody, myself included, was against it,
so my guess is that all of my party agrees that the DM CAN take away
players' agency under the right circumstances. That said, none of us
has any idea how to make the roll, but that wasn't a problem since I
decided to do what she was asking as it was perfectly in line with my
character's behaviour.
You were interested in a mechanical answer. You now have one.
I still recommend against it since roll playing this will rob both you and the other character of some role playing opportunity. If there is a successful seduction / persuasion check made, I further recommend that the two characters involved "fade to black" as they wander off to - off screen - resolve that interaction. The DM has other players, and needs to return the spotlight to the other PCs.
There are other RPG's that better handle this kind of interaction and play style, if your table is generally comfortable with role playing that kind of interaction.
I also suggest that you review this answer about sexytimes RPG approaches.
1What was that can of worms we were talking about?
The can of worms is the risks to the social contract that exists at a table, be it formal, informal, or a bit of both. Any player may have a certain limit, along the lines of "I don't want to have XYZ in games I play" where XYZ is anything that makes the player uncomfortable or get grossed out. (thanks, @Trish) A handy tool for any table is the X-card. I suggest that your group consider using it just in case this idea that you have takes you to some awkward or uncomfortable places.