Contains coNE, just as NEXP/poly contains coNEXP.
Has roughly the same relationship to E as PH does to P.
More formally, EH is defined as the union of E, NE, NENP, NE with Σ2P oracle, and so on.
See [Har87] for more information.
If coNP is contained in AM[polylog], then EH collapses to S2-EXP•PNP [SS04] and indeed AMEXP [PV04].
On the other hand, coNE is contained in NE/poly, so perhaps it wouldn't be so surprising if NE collapses.
There exists an oracle relative to which EH does not contain SEH [Hem89]. EH and SEH are incomparable for all anyone knows.