No.
Character Advancement
When adding new levels of an existing class or adding levels of a new
class (see Multiclassing, below), make sure to take the following
steps in order. First, select your new class level. You must be able
to qualify for this level before any of the following adjustments are
made. Second, apply any ability score increases due to gaining a
level. Third, integrate all of the level's class abilities and then
roll for additional hit points. Finally, add new skills and feats.
The order of how you level matters. You're allowed take a feat the same level you gain the prerequisites only because gaining feats is the last thing you do. If the prerequisite is a skill, spell, or ability, you already have it by the time you take the feat.
Of course, if the prerequisite is another feat, you must already have the prerequisite feat before you can take the desired feat. However, since character advancement is an ordered process, if you gain multiple feats in a level1, you can choose to take the prerequisite feat first. You can now take the desired feat.
In your example, you can take neither feat first since neither feat has its prerequisite met.
1 Class abilities that give bonus feats give them at the time you normally add new feats (after skill points). It's unclear if you choose your standard feat before your bonus feat or vice versa, but I would generally allow players to select feats in any order if it matters.
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.
Best Answer
James Jacobs says a bonus feat ignores prerequisites
The bard archetype dervish of dawn (né Inner Sea Magic's dawnflower dervish) does get the feat Dervish Dance at level 1 and can't meet the prerequisites. According to this May 2012 forum post from Paizo creative director James Jacobs, that's okay:
Presumably, Jacobs means a lone, named feat, not, like, all feats from a list or something. A fighter, for instance, still must meet prerequisites for fighter bonus feats, but a dervish of dawn needn't meet the prerequisites for the feat Dervish Dance.