Do you actually have to move with a Bull Rush target to move them more than 5 feet

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.

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

• from 30ft away?

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.

• 10ft reach weapon

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.