Would an Alchemist/Paladin be able to use his Smite Evil bonuses on bombs? Looking at the rules I can't see any reason why it shouldn't work. The rules refer to attack rolls and damage rolls (which I'd presume must result from said attack rolls), but they don't say that you can't use a weapon-like effect or ability instead of an actual weapon. The Bard's Inspire Courage ability is much more specific about "attack and weapon damage" rolls, but that language is lacking in Smite Evil.
I think you'd definitely get your Charisma modifier as a bonus to the bomb attacks. I believe that you'd also get your Paladin level as a bonus to bomb damage, doing the same type of damage as the bomb (fire by default). I think that the bombs would also bypass DR although that's not very helpful in this case since it is energy resistance which stops bomb damage, not DR.
In addition, while smite evil is in effect, the paladin gains a deflection bonus equal to her Charisma modifier (if any) to her AC against attacks made by the target of the smite. If the paladin targets a creature that is not evil, the smite is wasted with no effect.
The smite evil effect remains until the target of the smite is dead or the next time the paladin rests and regains her uses of this ability. At 4th level, and at every three levels thereafter, the paladin may smite evil one additional time per day, as indicated on Table: Paladin, to a maximum of seven times per day at 19th level.
Sure, you can do that, just as you can smite with a bow and arrow. However, the smite does nothing for splash damage because there is no roll for damage.
The splash damage for bombs is a little weird since it is based off of the minimum damage for the bomb. Since smite evil adds to "damage rolls" and you're not rolling I'd tend to agree that the bonus damage wouldn't apply. Even if it did it would only work against the target of the smite though (in case the bomb missed but landed near the target).
On the other hand, I'd say that the bonus damage from smite evil would still apply to Small shurikens even though you don't have to roll for damage since they just do 1. This makes me think of building a halfling Ninja/Paladin who destroys evil using a flurry of stars. It would be kind of odd if Medium paladins could smite with shuriken but Small ones couldn't, so I think it should work.
I don't think most DMs would analyze smite evil using a "normal" weapon that much though, so I doubt the Ninja/Paladin would have a problem. Does anybody else have an opinion regarding whether smite evil and bombs should work together per the rules though? Since bombs are a weapon-like supernatural ability rather than a normal weapon I could imagine many DMs balking at the idea of them working with smite. The name of the ability makes me think of a hefty melee weapon, but (as MurphysParadox points out) it clearly works with arrows, so why not bombs?
Most damage bonuses don't apply to the splash, so I wouldn't think a Paladin's smite or a Ranger's favored enemy bonus would apply.
What are some of those "most kind of damage bonuses" that don't apply? Othen than precision damage, every kind of damage bonus can be applied.