I think you can stop after 22 pt buy IMO. Consider the average level 1 monster stats: AC 12, 15 HP. If you pick one of any number of Full BAB builds that use the dog as an Animal Companion or Mount you're looking at moving half it's speed (20') and being able to make a ranged attack or a full 40' and still being able to engage in melee.
If you pick a Halfling, at level 1 with a 22 pt buy, you're likely looking at a Str 12, Dex 18 or thereabouts. Using a Halfling slingstaff that's a melee +3 (1d4+1)/ranged +6 (1d6+1) before feats, class abilities, etc. Plus your dog is making a melee (bite) +3 (1d4+1) attack as well.
Right out the gate you've got 60% chance to hit on both melee attacks, each of which deals an average of 3.5 damage. That's BEFORE feats, class abilities, etc. and it begins at level 1. From there you'll only continue to stack bonuses on top of that.
On the other hand if you attack from range and then have your mount move up into melee you'd have a 75% chance of dealing 4.5 damage to add to your dog's bite again, before other modifiers. Even if you only got 1 feat, taking Weapon Focus: slingstaff grants you a +1 on all your Halfling PC's attacks with the weapon.
Now remember: you're part of a party of other PCs. This means the expectation of your character in a CR 1 fight is to be able to contribute about 3.75 HP damage, or 1/4 the monster's 15 HP. You're surpassing that with a 22 pt buy with a single sling bullet, with just your stats.
So... your dog, you, the choice of a Halfling, and wielding a sling staff is what I'd suggest. Everything beyond that is icing on the cake.
Feats to optimize the weapon would be Weapon Focus/Specialization, Point Blank Shot, Precise Shot, Slipslinger Style, Rapid Shot and Halfling Slinger. Beyond that, there's lots of ways you could go to specialize.
With a Hunter, Inquisitor, or other build that grants free Teamwork feats, consider going Broken Wing Gambit/Pack Flanking/Paired Opportunists so that every round in melee when your foe(s) take the bait you and your mount and making an extra AoO.
On the other hand you could take the Mounted Combat tree while your dog builds as a Bodyguard archetype. Going this route, eventually far down the line your dog is setting up a Combat Patrol and making AoO's on foes getting too close while you're flinging sling bullets with little or no penalty no matter how far/fast your mount moves, while also making Ride checks to make sure the animal takes less damage.
One of the things in the OP was: being able to do something out of combat. With a 22 pt buy a Halfling could also have a decent Cha. if you instead focused some feats/teamwork feats on Intimidating, you could eventually intimidate up to Large sized creatures with a nice bonus. Another angle to go would be to capitalize on the high Dex you've already given the PC and be a backup Stealth PC. There's a couple traits that give it to you as a Class skill if you don't already get it.
The OP defines the 4 things you'd like to do to be the "best" with the build, but looking at the average stats for monsters, I still contend that the 22 pt build effectively guarantees a decent, if not great PC regardless of many build choices.
