Angel vs. Buffy
Once upon a time this would have been a lock for Buffy. However, since Angel got his own show he's really stepped up his game. He seems to pull off fancier moves than Buffy ever did.
Buffy seems to have faced stronger foes than Angel, but this seems mostly to apply to Big Bad types. On the whole, Angel seems to face powerful foes more often, and greater numbers of them than Buffy did.
Buffy may have beaten Angel in the past, but it's been what, six years since they last fought? I think that although Buffy was more powerful, Angel has made greater progress as far as improving his fighting techniques. These days I think he might win if they fought.
Angel vs. Buffy... Angel
Wesley vs. Giles
They're both mainly bookish types. A battle of wits between these two would be more interesting than a straight out fight, but since that's not the case...
As far as fisticuffs go, I think Giles has the edge. He seems to have more meat on his bones. Neither seems to be obviously superior as far as technique goes, so in the end I believe that Giles would win simply due to him being able to hit harder.
It would, however, never come down to a fist fight. Giles once said that his tweed was better than Kevlar... but he was wrong. Wesley has embraced the use of guns, and against humans they work quite well. Wes would be the clear victor here simply due to the tools he uses.
Wesley vs. Giles... Wesley
Fred vs. Willow
Willow has magic on her side. In a battle Will has the obvious upper hand. However, Fred has her own advantages these days. Fred/Illyria seems quite potent.
So it all comes down to what trumps what. Is magic better than ancient demon powers? The thing is, in the Buffyverse, magic is generally a plot device. It is a means to an end. It does what ever best suits the story and does not do whatever what ever detracts from it. Illyria's powers seem to still be a mystery.
Essentially we end up with one character having powers which aren't in any way quantifiable, and one whose powers have yet to be fully explored. In the end all we have to judge by is their history and performance. In which case I think Illyria wins.
Fred vs. Willow... Fred
Gunn vs. Xander
Xander has lost all his former military knowledge. He really doesn't have the skills or abilities to win this fight.
Gunn is a fighter. He has always been said to be incredibly strong for a human. Gunn is a fighter.
If the question was who was braver, I think Xander would beat almost anyone from either team. In a fight however Gunn would pummel him. Unless there's a wreaking ball nearby for him to use, Xander is going down.
Gunn vs. Xander... Gunn
Harmony Vs. Anya
As a vengeance demon Anya was a contender. She held her own against Buffy. Anya ended as a human though. She may have had some skills, but combat was never really her area. In the end it doesn't really seem t matter though, since Anya is already dead.
Harm is a twit. Plain and simple. She also can't really fight that well. She is however super human and that has to count for something.
Human Harmony vs human Anya, Anya would win. Vengeance demon Anya vs vampire Harmony Anya would win. Human Anya vs vampire Harmony I think Harmony comes out ahead (but still has to work for the victory). In the end though, the competition would be between vampire Harmony and corpse Anya, in which case Harmony would win without even showing up.
Harmony vs. Anya... Harmony
Lorne vs. Dawn
Dawn... has... um... well, she... uh... she's got the ability to tell Lorne that "If you hurt me my big sister will come and beat you up!"
Lorne has an ear piercing scream. He has some level of invulnerability (he's had his freaking head cut off for Pete's sake... how many other Buffyverse characters could survive that?). He comes from a race of warrior demons... which pretty much has to have some kind of perks. He also has a killer wardrobe. I mean, those suits rock!
Neither is a great fighter, but Lorne is bigger. He'd just sit on Dawn until she said 'uncle' all the while entertaining her with a chipper little song.
Lorne vs. Dawn... Lorne
Angel's Avengers vs. The Scooby Gang
In a free-for-all battle royal there are two possible outcomes that I foresee.
Outcome one, is that the Scoobies, now possessing an army of Slayers, stomp the Fang Gang with ease.
Outcome two has the Wolfram & Hart crew forgoing the whole 'mano-a-mano' bit and frying all the Scoobies with their cool death ray satellites. You know, I wish I had death ray satellites.
Since the satellites could be implemented without ever having to face their opponents, the Fang Gang gets a preemptive victory by frying their foes without ever having to put themselves in danger.
Angel's Avengers vs. The Scooby Gang... Angel's Avengers