We have Half-Elves, Half-Orcs, and (I hope I remember it right) in the Kaer Maga Handbook they even mention that mixing halflings and humans often ends up in handicapped children. But I have never heard of Half-Dwarves in Golarion. Is there a reason why they do not exist in Golarion? Or do they exist and I have simply not found the source yet?
Golarion dwarves cannot breed with non-supernatural races
Player Companion: Bastards of Golarion has a sidebar discussing this. For whatever reason, Pathfinder's developers have decided that gnomes, halflings, and dwarves cannot have crossbreeds with other PC races. It's detailed in a sidebar on page 33. I'm not sure the design and writing reason, but the in-universe reason is as follows:
WHY ARE THERE NO HALF-DWARVES?
Although humans can have children with many other races, there are limits to this gift. Half-dwarves, half-gnomes, half-halflings, and numerous other combinations are all but unheard of on Golarion. Biological incompatibility is the first and foremost reason that such half-races cannot exist. Simply put, dwarves, gnomes, and others just aren’t compatible with other races, even humans.
In a realm shrouded in magic, it would be foolish to assume no spell in the known multiverse could produce a viable child between a dwarf and a humanoid of another race. Indeed, dwarven aasimars and gnome tieflings are known to exist, arising through the influence of outsiders or because of magical anomalies understood by few. A miracle or wish spell could likewise result in the birth of a half-dwarf, though the individuals able or willing to practice such spellcraft are few and far between.
From a gaming history standpoint, like a lot of things this can be traced back to Tolkien. The Lord of the Rings has both half-elves and half-orcs, but no half-dwarves or half-hobbits (the people of Gondor refer to hobbits as "halflings", hence the D&D name, but that's because they're half the size of a human, not because they're the product of interbreeding).
Pathfinder inherits the player races of D&D 3.5, since much of the setting was drawn from D&D games that the designers ran, and because one of the design goals was to be backwards-compatible with D&D 3.5.
Originally, the Derro were supposed to be half-dwarves/half-human, as illustrated by this post from James Jacobs (Paizo's Creative Direction) from 2007, and again on this thread, illustrating his love for derros.
Aparently, the source of that is much older than Golarion. Back in the Monster Mannual for 3.x, Derros used to be half-dwarf/half-human. On D&D 2nd edition's Scarlet Brotherhood, a regional sourcebook for the Greyhawk campaign setting written by Sean K. Reynolds (ex-developer from Paizo) in 1999, also claimed that they were dwarf and human hybrids created by wizards that wanted a race of thralls completely loyal to them (the Derro, here, being a failure on that objective).
The source of all this might be the 2nd edition's Monster Mannual II (1983, by Gary Gygax) that stated that derro were possibly half-human/half-dwarf hybrids.
The derro are a degenerate race of dwarf-like stature, possibly a cross between evil humans and dwarves. They inhabit the great subterranean realms common to drow and their ilk (kuo-toans, mind flayers, troglodytes, etc.).
Derroes are very much like humans, only shorter and slightly more muscular in proportion to their height. Their features tend towards grossness. Their hair is pale tanoryellow, their skin isverywhitewith a bluish undertone, and their eyes are very large.
Back when the Curse of the Crimson Throne was originally published (2008), all references to Derros used the 3.5's Monster Manual and nothing in the book mentioned anything about their origin or what they actually are. It's safe to assume their new identity with new flavor wasn't set in stone yet.
Their new flavor and origin in Golarion was actually revealed in the Into the Darklands campaign setting book (later that same year):
The origins of the derro are murky at best, but study of ancient orc carvings and dwarven texts reveals a disturbing truth—the derro existed long before either of those races rose to prominence in Nar-Voth tens of thousands of years ago. (...) In fact, the first derro are the descendants of a much more benign stone-loving fey race known as the pech. After the fall of the Vault Keepers, one tribe of pech chose to flee upward through the Darklands, and when they reached the surface world they realized they had gone as far as they could go.
You asked why we don't have half-dwarves, since we have half-orcs and half-elves. When asked why we got half-orcs (2008), James Jacobs answered this:
We're going with half-orcs for two reasons.
1) Half-orcs have been a player choice for race since first edition, and as the previous poster mentions, we're trying to retain the feel of D&D as much as possible in the Pathfinder RPG.
2) Orcs are monsters in Pathfinder, not heroes. There are several other great games that present orcs as heroes, but that's not the type of game we're trying to present the baseline for with the Pathfinder RPG. The core rules are meant to steer players toward playing the more humanlike races for a reason; because the main world those rules will support, Golarion, is a world built with those races in mind. In Golarion, the orcs are ravenous, brutal, savage monsters; an orc PC wouldn't fit well into the setting as a result unless the campaign he was playing in was an all-orc game.
That all said... this doesn't mean that you can't choose to play an orc anyway. Nor does it mean we'll NEVER present the orc as a PC race in an expansion to the rules a few years down the road. It's just not one of the baseline choices we want to present for the core game.
In closing... you're right about there not being many half-orcs in the world. There aren't many half-elves either. But both races make GREAT choices for PCs, since this lets a PC play the "underdog" or the character who has no real home to call his own. Also, keep in mind that in any one campaign, you've only got around 5 PCs at a time, so even if ALL of them are half-orcs, that's still only 5 people overall; half-orcs can still be rare and be a PC race without making the race common.
Half-dwarves, aside from Darksun campaign setting and the Derro from Greyhawk, have never been a popular player choice, unlike half-orcs and half-elves.