How things work in Pathfinder
- Haste can target any creature. That's what haste's target line is for. It says:
Targets one creature/level, no two of which can be more than 30 ft. apart
All animals are creatures, so of course you can target them with this spell. Since nothing in the spell cares what kind of creature the target are, it affects them no differently than it would any other creature.
Note that the haste being mythic has nothing to do with this. All the mythic part does is give the targets extra actions in addition to extra movement.
- Ultimate Combat gives stats for carts. As you can see, both the maximum speed and maximum safe acceleration limit are based off of the speed of the pulling creatures:
Maximum Speed twice the speed of the pulling creature(s) –10 ft.; Acceleration the speed of the pulling creature(s) –5 ft.
The horses pulling your carts thus ordinarily accelerate 15 feet per round to move the carts up to 30 ft per round at maximum speed. While subject to haste, they have a speed of 50 feet instead of 20 and thus can move the cart up to 90 feet per round with an acceleration of 45 feet. While mythic haste gives them an extra move action, the cart's movement is not dependent upon any actions from the pulling creatures (but it is dependent on the driver's actions). So the cart can go much faster, but can't benefit from the mythic part of the Haste spell.
- Occupants all move with the vehicle. There's no roll involved for that; only if the vehicle crashes or is rammed or something would the occupants need to worry about death via being thrown from the vehicle:
Sudden Stops: When a vehicle comes to a sudden stop—its movement is reduced to 0 in some way other than the driver using a drive action to slow the vehicle—both creatures and items on the vehicle are violently pushed toward the vehicle’s forward facing a number of squares equal to 1/2 the vehicle’s current speed before it came to the sudden stop. This movement does not provoke attacks of opportunity. At the end of this movement, creatures and objects take 1d6 points of damage, and creatures must succeed at a DC 20 Reflex saving throw or be knocked prone. If the movement pushes creatures or objects into solid objects, that creature or object takes an additional 1d6 points of damage for each 5-foot square the push was reduced by the solid object. For instance, if a vehicle with a movement of 60 feet makes a sudden stop due to hitting a brick wall, its driver is thrown 30 feet toward the brick wall. If the brick wall was only 5 feet away from the driver at the point of impact, the driver moves forward 5 feet, hits the wall, and takes 5d6 points of damage. She then takes the original 1d6 points of damage, after which she makes a Reflex saving throw to see if she falls prone for the sudden stop.
So, speeding up the carts makes things more dangerous if you crash or something, but won't hurt the people if you make it safely to your destination.
- Horses go faster than that. It sounds like your horses are encumbered and fatigued, but not zombified nor caltropped.
A typical heavy horse has a speed of 50 feet. Encumbered, that speed drops to 35 ft. Fatigue does not affect speed, but exhaustion halves it to 25 ft. An exhausted, encumbered draft horse has a speed of 20 feet.
A typical pair of draft horses can pull a 24,000 lb load with a suitable carrying apparatus. It is unclear whether or not doing so encumbers them, but it's fair to rule that it does. If it does, pulling 'merely' a 7,980 lb load should not (that's taking the light load limit, rather than the maximum load, and applying the multiplier for dragging etc). Thus, your horses probably shouldn't be encumbered unless you have at least, like, 20 guards per cart.
A typical pair of draft horses can pull a cart for just under 9 hours a day before becoming fatigued. Fatigued horses take 1d6 lethal damage as soon as they become fatigued (this, in fact, is what renders them fatigued) and each hour of riding thereafter, which will kill the average draft horse pretty quick, with their mere 19 hit points. You've only ridden the horses for much less than 8 hours, so unless you've forced them to hustle (which you aren't normally able to do in a cart) they shouldn't be fatigued.
Carts in Pathfinder are also weirdly weak and can only hold 300 pounds of cargo, though they can also hold up to two medium creatures and all of their gear. You can't pile more than two guards into such carts as cargo, but you could totally do so with a light wagon, so that's not that difficult to explain.
In any case, given the anamolous behaviour noted above, I suspect your DM is not using Pathfinder for this particular encounter, in which case you would need to find out what system your DM is using to be able to answer those questions (and just in general, if you want to be able to predict the probable effects of your actions).