I strongly disagree with Phill.Zit's reading here. Your own quote says it all:

the weapon’s bonuses apply to the roll.

It goes on to clarify ('therefore…') this would require a finesse weapon. But bows use Dex as their bonus despite not being finesse weapons, so should use Dex for these checks.

In fact, I'll go further and say you have to use Dex. Finesse weapons give you the choice, whereas bows do not.

And note the wording on the types of combat maneuvers it applies to. Normally with a bull-rush or grapple

you're [not] actually using a weapon

so you'd be forced to use Str. But again, here you are using a bow so must use Dex.

You're Correct

The discovery Explosive Missile says

As a standard action, the alchemist can infuse a single arrow, crossbow bolt, or one-handed firearm bullet with the power of his bomb, load the ammunition, and shoot the ranged weapon. He must be proficient with the weapon in order to accomplish this. When the infused ammunition hits its target, it deals damage normally and detonates as if the alchemist had thrown the bomb at the target.

The word can applies to all three of infuse, load, and shoot, which collectively take but a single standard action.

It's a matter of character and campaign whether this discovery is valuable; that is, to use it the character must A) usually make a ranged attack instead of a ranged touch attack, and B) factor in the additional expense, inconvenience, and weight of the weapon and its ammunition.

The discovery Explosive Missile is remarkably evenhanded. The discovery changes the typical crossbow-toting alchemist's bombs from bombs that're ranged touch attacks to bolts that're ranged attacks that also inflict bomb damage. Further, the discovery permits the alchemist to quickly load and fire that crossbow. The game realized the alchemist was already sacrificing a significant advantage by having the alchemist make a ranged attack instead of a ranged touch attack and compensated for that by permitting the alchemist--in some unexplained, maybe magical way--to reload and fire even a heavy crossbow as a standard action if the alchemist expends one of his bombs per day. That's a superior balancing act.