for example, can an interposing hand be affected by haste?
also, if the caster has a false life in effect, is this taken into account then the hand's hit points are determined?
As long as the target satisfies any target conditions on the spell or effect, yes.
As to fhe first question, haste targets creatures, is the hand a creature? Doesn't look like to me.
As far as the second question, temp hp are in addition to your current hp total, not part of it.
Summonings from conjurations usually yes. Force constructs brought about by evocations like your hand spell, not so much.
As the others pointed already out, the target conditions must be fulfilled. Temporary hp don't count for such calculations, but I see where you come from :)
On a sidenote: haste makes no sense on the interposing hand, even if it would work. If it doesn't stick to it's target, it cannot block it and it cannot be in two places at once. It also has no attacks of its own.