1.
2. Unless they use an IGZO display, which would reduce power consumption and heat substantially. In combination with a larger battery (I'd say around 25-30w, which is at best +30g weight) battery shouldn't be negatively impacted and weight is largely the same.
3. "Pixel doubling" is stretching which is ugly as all hell. You're talking about scaling.
4. The next step in resolution is 3072x2304, not 4096x3072. Try not to think of the iPad 3+'s resolution as 2048x1536, rather 1024x768 with a scaling factor of two. Anything that's a multiple of 1024x768 is possible while maintaining app compatibility. At 3072x2304 the scaling factor becomes three. I could explain in more detail if you wanted.
5. It would only break app compatibility as much as the iPhone 5 did, which really wasn't much. However I don't believe the iPad needs an aspect ratio change, so there's no reason for them to do it at all.