I'm not sure about whether she's flying or not. As far as I'm concerned, the best evidence that she can fly comes from the part where she swoops in and knocks Doomsday's leg out from under him with her shield. She appears to skim along the ground in a very unnatural way. For my money, it looks like she was somehow defying gravity there. Well, that or she had rollers in her boots
As far as power levels go? Diana shows four strength displays and several durability displays.
For strength?
1.) Doomsday throws a car at her and she slices it in half easily. Some might argue that it was the sword doing the cutting and not Diana. This is incorrect. Physics states that it doesn't matter how sharp your blade is, if you can't apply the pressure needed to break the surface tension and draw the blade through the substance in question you're not going to cut it. All those comic book scenes where Wolverine cuts Hulk with his claws? Nonsense. It doesn't matter how sharp those claws are, Wolverine needs to be strong enough to pull them through Hulk's flesh. Hulk's flesh laughs at bullets, artillery shells, and missiles: all things that can do far more damage than Wolverine's muscles. Hence? Wolverine can't cut Hulk no matter how sharp is claws are.
Give Diana's sword to Batman and invite HIM to cut a car in half, and he won't be able to do it. At least not in one clean swipe like Diana did. It was Diana's strength that allowed her to cut that car in half.
2.) She blocked Doomsday's punch with her sword. Physics says that Diana would have to exert equal or greater force against Doomsday's arm in order to stop it like that. Hence? She exerted enough force to stop a punch from a creature that was knocking Superman around with ease. There is no reason to assume Doomsday threw a weaker punch at Diana. Doomsday is a rabid animal. Pretty much every strike he throws can be assumed to be as strong as he can manage because he is ALWAYS going for the kill.
3.) She cut off Doomsday's hand. See Example 1. Doomsday's flesh survived a nuclear blast, bullets from Batman's plane, and multiple punches from Superman. For Diana to cut through flesh that tough (again, with a single clean strike) would have required her to be extremely strong indeed.
Once again, Batman could not have achieved the same feat
4.) And she held Doomsday immobile with the Lasso for several seconds. Sure, she lost her grip eventually, but Doomsday was getting stronger by the second at that point. Some might argue that it was the Lasso holding him, but that makes no sense. Why would Diana have to brace herself against the ground and haul back on the rope if the rope was doing all the work of keeping him immobile?
Durability? She took several punches from Doomsday and SMILED at him after one of them. Some have argued that Diana didn't survive those impacts, it was her shield doing all the work. This is also impossible. At one point, we see Doomsday punch her out of midair as she leapt (flew?) at him. In that situation, her shield would've been no defense at all. With her feet off the ground, the shield would not be able to brace against anything. Even if Doomsday DID punch the shield, it wouldn't have been able to properly absorb the impact. If anything, the punch would've pushed the shield back at Diana, causing her to smack herself in the face with her own shield.
We also saw Diana take a heat vision blast from Doomsday immediately after she cut off his hand. She did not have her shield at that time. That was after she lost it in the fight. So unless Diana got her bracelets up in time to intercept the blast, then she took that blast full on and wasn't hurt by it.
Speed is a little harder to gauge. Diana definitely came flying at Doomsday at pretty extreme speeds. When she rushed at him at one point, you can hear a faint "Whoosh" sound and a slight blurring around her. In movies and TV, these effects almost always denote some degree of superhuman speed is at work. So it's pretty obvious that she's fast. How fast? Unknown. We may have to wait until the JL movie so we can see how her speed compares to Flash's speed.
The important thing about this fight? Diana went toe-to-toe with a monster that could survive nukes and an onslaught from Superman. At no point did she appear hurt or even tired from her exertions. At the end of the fight, she wasn't even dirty or cut or bruised. She looked for all the world like she could've gone another ten rounds with Doomsday, and it's important to remember that this may very well have been the first major fight she'd been in for the better part of a century. This was a "rusty" Diana we saw. In future movies, when she will no doubt be getting into even better fighting shape? She's probably going to be even MORE formidable.