Exactly what here is referred to be "faster than light?"
By fast than light it means that if you put a light beam to your left and to your right and you ball take off like racers running towards a take across a finish line that you get to the finish line first.
And how does this work? Are you clumping together a punch of exotic matter at the front/rear and that bends spacetime?
There are different warps drives one of them compresses the space in front of you but presses it out like a pancake so the volume doesn't change, others literally do contract the space in front of you. And yes you do it will exotic matter because spacetime doesn't normally do that on its own.
What kind of energy levels are we talking about here?
Depends how big you want it to be, the issue is the density of energy. Imagine the biggest density you've seen. Now imagine something bigger. Bow imagine something even bigger, an infinite density, now imagine something even bigger still, a negative energy density.
And don't forget that a particular kind of energy density moves according to its own equations so you might have to arrange in advance to get it to show up to the place you want at the time you want.
Can we even produce a such power needed to bend spacetime that much today, with current technology?
If you tried to use, for instance Casmir energy densities then the negative energy densities get larger the closer the plates are together, so maybe you could make a really small one that doesn't go very far since the plates are so close together. I didn't think anyone wanted that so I don't know of anyone trying to build one.
Are the any rest products (exhaust/waste), and how are those handled?
When the metric was written down it was just in the style of "what if there was a spacetime like this what would the metric and hence the stress-energy be like." But the reality is that the stress-energy of known forms of matter have their own evolution equations, dust moves on geodesics, gases follow pressure gradients, charged particles are pushed by electromagnetic fields, etc. No investigation has been made into how to make any of these forms of matter move the right way because they wouldn't work anyway.
If you have some negative energy density you can ask how to make it move. For instance that Casmir energy has to have more positive energy density in the plates than the negative energy density in between the plates so if you want it to move fast you have to make those plates move fast and if you try to accelerate the plates then the Physics is different.
The point is, a constant input of energy is needed for a traditional rocket to accelerate. How is this energy provided in a warp drive, and in what form?
If you have a payload that is not exotic and you are also transporting exotic matter it isn't obvious that you need any fuel since for all we know you are having equal amounts of positive and negative energy arrive and maybe the same for momentum.
The real problems are that space isn't empty and this is going to get in your way. So badly that maybe you just can't go FTL even with exotic matter. And this isn't just space debris, even the quantum vacuum might be a problem. And the original drive couldn't turn and didn't start up or slow down, so even if the original drive was possible if you can't get started that might be a problem.
So how much field to get started or to steer? the original drive had no steering and no start up or slow down. Maybe that can't be done.
Does it have to constantly put in more energy to maintain the curvature, or can it just "cruise along" the bubble?
The part in the middle definitely just cruises, they could be at rest from their own point of view. But the exotic matter has to follow it's own laws which depend on the particular kind of exotic matter.
If a light beam was sent alongside the warp drive, which would arrive first as measured by observers on the Earth, in the warp drive and at the destination?
The warp drive arrives first assuming the are far enough away from each to not affect each other.