Assuming they had all their gear and stuff, for the first one, well I don't know them all, but here's how it would turn out.
Batman would lose because, "No guns, no killing," automatically makes him fail. Indiana Jones kinda sucks, I mean, a whip? Come on. He could kill William Wallace though because, well, you remember the scene where he shoots the guy who has a knife. Hit Girl knows her karate shit but she just isn't strong enough to go against someone with equal martial arts skill. Therefore, Beatrix would totally take her out. Bear Grylls might think he's the ultimate survivor, but Alice lives in a wasteland that makes his adventures look like picnics. She'd kill him so fast. Anyway I'll do the rest of the explanations for Round 2.
Batman -- Critical Failure
Indiana Jones > William Wallace
Beatrix Kiddo > Hit Girl
Aragorn > Xena
Leonidas > Achilles
Alice > Bear Grylls
Sarah Connor > Katniss
Maximus > Rambo
Ripley > Everyone Else
That leaves us with: Indiana Jones, Beatrix Kiddo, Aragorn, Leonidas, Sarah Connor, Maximus, Alice, and Ripley.
In a fight to the death, I'd choose Maximus over Beatrix. Sure she knows her ninja arts and shit, but Maximus is much more cunning. For example, in the second Kill Bill movie she nearly dies because she opened the front door of the trailer of the brother her arch nemesis. I mean, come on. At least scout the place out first. Maximus would never have made such a fuck-up.
Maximus > Beatrix
Leonidas versus Aragorn is a hard one, but I'll side with the Spartan for this. Aragorn is a good ranger but Leonidas was raised to fight since he was a young boy.
Leonidas > Aragorn
Sarah Connor versus Indiana Jones is next, and here I have to side with Sarah. Fighting is something Indy does, but he's never had to kill a time-traveling cybernetic killing machine before. She also knows everything about weapons and explosives. Indy just doesn't have the experience.
Sarah Connor > Indiana Jones
Speaking of messed up experiences, the next match-up is Alice versus Ripley. Both of them come from fucked up situations in the future involving high technology, but tell me, which do you think is worse, an army of zombies or an army of Xenomorphs? I'd choose the zombies any day, but Ripley never had that choice. Nonetheless, she came out on top against those monsters. She walked right in the front door of the Alien Queen's lair, took one look at her eggs, and fucking torched the place, just like that. Alice has intense combat experience, but only one of her opponents was even close to the Queen that Ripley took down. You might say, "Oh but Alice has like a hundred clones of herself and she is a psychic." Well, if you watched Alien Resurrection, you'd know that Ripley also gets cloned, but in even more jacked up ways. She becomes part-alien herself and inherits a lot of their abilities and senses. Ripley wins here.
Ripley > Alice
Maximus, Leonidas, Sarah Connor, and Ripley.
Let's do the heroes of the past first, Maximus versus Leonidas. I think King Leonidas is the fucking shit, but I'm going to give this one to Maximus anyway. Maximus is probably a better hand-to-hand fighter. Leonidas obviously knows what he's doing, but the kinds of opponents he faced were all the same -- enemy soldiers. Maximus had to fight in all sorts of situations when he and his companions were totally screwed. He even killed some Pretorians, the most elite soldiers in Rome which eventually took over Greece I might add. Leonidas went up against great odds, but they weren't the same kinds of individual versus individual combat situations that Maximus had to deal with.
Maximus > Leonidas
And the next is Sarah Connor versus Ripley. This is probably the toughest one. Since Ripley isn't against a psychic here, let's assume she's back as a normal human like at the end of Aliens before those other directors fucked up the story. Sarah and Ripley both had to fight against some of the most evil beings ever conceived by Hollywood. The Xenomorph, a perfect killer, its lifecycle depends on killing humans for reproduction in an incredibly violent way. It has acid for blood, is stealthy, and two sets of teeth what the fuck. Then of course, Sarah dealt with two Terminators, mechanical monsters from the future which traveled back in time with the specific mission of killing her and her son. For this, although Ripley is probably equally badass and just as tough, I'm choosing Sarah Connor. Between the first and second Alien movies, Ripley only spends about 3 months awake, between her first and second fights with the Xenomorphs, most of her time was spent in hibernation, not training to become the best fighter ever. Sarah, on the other hand, had years of accumulating weapons and learning about war. Ripley showed that she could be a good commander, but she just never got the same kind of training that Sarah got for herself. I'll choose Sarah here.
Sarah Connor > Ripley
Maximus versus Sarah Connor
The ultimate fighter versus a woman who has become as cold and brutal as the machines she destroyed. Sarah uses weapons Maximus could never envision, but she also spent months at the Pescadero training to fight, getting stronger and stronger every day, and years before that learning how to be a warrior in Mexico. Maximus spent years in the gladiator camps, forced to kill people, against incredible odds. Giant Cats, chariots, and scores of enemy soldiers all fell before him. They both know what they're doing. But in the end, Maximus doesn't have a SPAS-12. There is not much that Roman body armour can do to stop one of those. :/ Sarah Connor takes all.
"Each night alone I dream, that I'm a rebel Roller Queen‼
I'll be a star that shines, I can make the whole world mine‼"