In many ways, they behave like spells in potion form, and as such their effects can be dispelled by effects like dispel magic using the alchemist’s level as the caster level. Unlike potions, though, extracts can have powerful effects and duplicate spells that a potion normally could not.
Potions are like spells cast upon the imbiber.
A creature that takes a direct hit from a healing bomb is healed as if she had imbibed the infusion or potion used to create the bomb.
I dunno. It's not definitive, but there's definitely an argument there.
Does an alchemist qualify for prestige classes that require you to be able to cast spells of x level? Does eschew materials apply to their extracts?
If so, yes it applies. If not, then no.
My take is that they do not qualify for prestige classes that require you to be able to cast spells, and that eschew materials doesn't work for them. Thus they wouldn't benefit from the healing domain.