What do we know about $text{P}^text{NE}$

I have a $text{NEXP}$-hard problem, that can be solved by a $text{NEXP}^text{NP}$ algorithm using a single oracle call.
So from Hemaspaandra we know it is in $text{P}^text{NE}$, giving us
$text{NEXP}$-completeness under Cook-reductions.

• Are there (more or less) natural problems complete for $text{P}^text{NE}$?
• What do we know about the space between $text{NEXP}$ and $text{P}^text{NE}$?
• Anything else that might help to sharpen this result (to Karp-reductions)?