No, without waiting until a much later level for improved critical you can't do this. At level 10, you can pick up Vital Strike (or level 9 if your GM will allow you to retrain your animal's feats) which is kinda-but-not-really similar--but if you want more than one attack Vital Strike is wasted as feat (a wolf works really well for this Weapon Focus, Improved Natural Attack, and Vital Strike then cast Strong Jaw and it works wonders).
Another option that doesn't really work is that animals can make use of Amulets of Mighty Fists which can be enchanted. However, keen is only applicable for piercing/slashing weapons and an Amulet of Mighty Fists can only be enchanted with enchants that are applicable for bludgeoning weapons.
At this point, it's up to your GM to allow you to have either a custom AoMF or some kind of "animal companion" weapon with keen (though even a single keen weapon will only work for one natural attack).
First of all: your reach is your weapon's reach. Different attack forms may have different reaches but the 'reach' listed for each creature size is just the default and is used for most creatures and most attacks. Wielding reach weapons is kinda like sprouting tentacles in that way; both of them give you reach longer than the 'normal' amount. You threaten area with each of your attack options individually. Everything else about Reposition is pretty clear, I think, so that one doesn't really need further clarifying.
For the rest of them:
This FAQ post clarifies the issues here, which are really reach related. Pathfinder does not do a good job of defining several fundamental game terms in-text, of which reach is one. 'Reach' doesn't just affect where you can attack from, it affects you ability to to just about anything at range. Abilities, like spells, that specify their own ranges overrule this because specific>general but in general you can only affect things within your reach.
Now, onto further detailed discussion of the consequences for each:
Bull Rush
As a standard action? Only if you can reach them. This will generally involve either being Colossal or being at least Huge and wielding a reach weapon. However, you can Bull Rush as part of a charge, so as long as your speed is at least 15 ft this is still doable as part of a full-round action.
Absolutely, and you do it via reach granted by the weapon so "keeping them at bay"/"pushing them back" is a totally valid way of describing that in the game world. Your character concept is solid here RAW.
- "Can I Bull Rush adjacent enemies back 5' so they are threatened by my hammer? Can I do this while unarmed at that range?"
Your unarmed strikes are an attack form you possess, which tracks reach separately from your weapon as discussed above. You can't bull rush an area you can't reach, but you can generally reach the area next to you (even while armed with a reach weapon). You will be doing this with your body, not your weapon, though, and may incur AoO/nonproficiency penalties as a result. You may instead be able to do this using your haft as an improvised quarterstaff given the poor reception of my answer here, which would negate any reach-weapon related issues you have at that range.
Dirty Trick
This one's special. It specifies both that it "covers any sort of situational attack that imposes a penalty on a foe for a short period of time" and "The GM is the arbiter of what can be accomplished with this maneuver" which seem to be indicating that this manuever would be getting a lot more free reign RAW in terms of what it can do. However, the text does specify that Dirty Trick can only be used "in melee", so I think this is still limited to reach in terms of range.
Drag
- so can I not use the hook on my pole arm to drag someone?
Drag just doesn't work RAW. It's not that you can't drag people via a reach weapon (you certainly can, according to the rules) nor that you can't drag people of a different size than you (which is even included in the 'tactics' section of some creatures), its just that the rules make no sense if you do anything but drag an adjacent Medium or Small sized creature and don't think too hard about the Pathfinder definition of a straight line. By "make no sense" I don't mean that the rules are confusing or that they don't correspond to some particular assertion of what physics are like 'in real life', I mean that they seem contradictory. You, apparently, both move the opponent 5 ft and cause them to occupy your former space, which in any case but the specific two cases in which this works, leads to your opponent being in two places at the same time and possibly you as well, while simultaneously not being in either place. I suppose you could run this like some kind of macroscopic quantum superposition effect, but I think it's much more reasonable to just accept that the designers messed up and come up with your own rules for dragging. You should definitely be able to do it with a pole-arm, just ignore the bit of text about moving into someone's space and the rules work much better (though the 'line' thing is still an issue).
Grapple
- under what circumstances can you do this from more than 5' away?
Whenever you have more than 5' of reach. Pretty simple, this one.
Best Answer
While there may be alternative ways to increase your threat range, none of these stack. Keen, Improved Critical, etc. are all very explicit about this. The feat itself states:
In fact I would argue that if there is any possible stacking of critical range increases it is merely an error in the wording and wouldn't allow it in my games.
So taking Improved Critical, if you can spare the feat and know which weapon you want to focus on, is the cheapest way to get the highest range possible.