1: Not anything exactly like that with RAW. So, up to the GM. A diplomacy check would be in order. The DC would be based off of how extreme a response you were trying to get (just trying to get them to smirk and talk to you, probably only a 5. Get someone to come over and have drinks, higher).
Circumstance bonuses should be used heavily, based off of roleplaying, dialogue, gifts (buying a drink) and what not. Circumstances might also make it harder.
This all really falls into GM fiat, but if you something a bit more mechanical, make a 'smitten' condition modeled off of shaken. PC makes a diplomacy check, the DC is 10 + target's HD, + it's cha modifier. On a success, they become 'smitten' for 10 minutes. Being smitten is effectively like being charmed (as the spell).
2. Pathfinder usually doesn't have ways for NPC's to ... socially compel (as opposed to magically) PCs to do stuff. I personally favor NPCs being able to affect PCs. It's a roleplaying game after all. Seems kind of silly that NPCs can't use intimidate or diplomacy to change a PC's attitudes. If you're like minded, use whatever rules you'd give to PCs.
3. Charisma doesn't necessarily equate to attractiveness. It is a pretty effective gauge of presence though.
But yeah, the closest thing you've got to a rating of someone's looks is their charisma, or perhaps their diplomacy modifier. You could roll off the 2 npcs on each other. Maybe they hit it off, or maybe only one does and the other one's just playin'.
It means they are attractive to the other tentacled horrors below the earth.
But then you have the problem of how one species of 18 CHA horrors sees a totally different species...
About the OP: Generally, the only reason I would want some attraction mechanic in the game is to have fun role-play. In that case It would be best to have some idea of what my character wants in a romantic partner and role-play it out.
The place I would really want rules is if a PC is being seduced unwillingly. For example, a courtesan is trying to lead a bodyguard away from the king he is guarding. In this case, the appropriate mechanic would be will saves. I'd set the DC at around 5 or 8 plus 1/2 the seducers HD plus CHA. I see it as the PC's sense of duty fighting the libido. I use 5 or 8 as the base because the effect is not magical and the seducer will get many attempts.
The reason diplomacy is bad for this is because it is very easy to get a diplomacy score high enough to instantly make a target "friendly." This takes choice away from the PCs in an unfair way and is not fun because it acts like dominate person and has no save...
There are no rules for attraction.
A high Cha can mean attractive or maybe they command attention in a confident way...
Really this requires the GM to know a lot about the NPC's personal life. Then it requires a lot of roleplay. Skill roles such as diplomacy can be useful if you want to impress her or get her to agree to something she would be willing to do anyway. But remember the skill toss is temporary, How you actually treat her is lasting.
Diplomacy is the easiest way to go I think.... There could be a ton of other factors involved, but I think Diplomacy should be at the heart of it.
*sigh* Must everything be done using dice mechanics? If it's going to further the plot or simply provide fun for the game, why not just wing it and make a fiat GM decision on whether or not there is attraction there? Not everything has to have a rule.
If it's a case of trying to vamp an NPC using sexual attraction as a tool, then Bluff would be the way to go. (You're feeding them a line of B.S. in order to get what you want.)
This question arose in-game: are there rules for determining in-game attraction?
Scenario 1: PC flirts with NPC. What does one roll to determine if the flirt successfully charmed the NPC?
Generally, Diplomacy is used to shift attitude. Physical attraction is of course a matter of personal taste. Sense Motive would be the go-to skill to determine if someone (and NPC) finds your PC attractive.
Scenario 2: Same as 1, but vice versa. What does an NPC/PC roll now?
Well, the GM's description of the NPC should be enough for you to determine if the NPC is 'attractive'. The NPC can make Diplomacy checks, but barring magic, a PC's response should not be determined by dice rolls. I'd recommend a successful Diplomacy roll to be described by the GM thusly: "The NPC seems to like/respect/admire your character." From there, Sense Motive would be used to determine their motivations; honest interest or something more selfish?
A failed Diplomacy roll would be described differently: "The NPC seems to be trying to talk you into something." Again, Sense Motive helps to determine what.
Scenario 3: Remove flirting from the equation and assume the two imaginary folks are in a tavern together, perhaps not even talking. How does one determine attraction? Just a straight up CHA check? I.E., "he/she has 18 CHA, so yeah, you find them very attractive."
Attraction can be completely nebulous. Assuming you want to attract someone, a straight CHA roll might be best. The DC of this check is probably directly related to the overall compatibility between the two. For two regular humans, DC10. For near species matches, DC15 (human/elf). For nominally compatible species (human/outsiders), DC20. For very strange pairings, DC25 (or higher). Generally permit PCs to be attracted to whomever they please, since they will anyways. You could still roll, of course, and describe the NPC as 'strangely alluring' to the PC; people do sometimes fall for people they normally would not at first blush.