You should be fine as long as you are consistent. If you use total pressure, as they do, then you also need to include the effects of the air pressure on top of the tire pushing down on it with 1 atmosphere of force!
As a thought experiment let's use some really fancy spherical-cow materials. Let's make a gigantic tire out of material that has negligible mass and put 100kN of force on it. If we pump the tires up to slightly above atmospheric pressure, say $P_{gague} = 0.00000001 kPa$. These are the flattest tires on the planet!
By your method, using just gauge pressures, you would see that these tires would flatten out like pancakes until their area was massive.
By using their method, converting to absolute pressures, $P_{total}=101.00000001 kPa$, so by $A=\frac{F}{P}$ you would get an area of 1 square meter or so. This would be an absolute minimum. No matter how flat your tires are, they would never take up more than a square meter to support 100kN. Something's wrong!
On the other hand, if you use all absolute pressures and you account for the fact that there's atmospheric pressure pushing down on the tires as well as the weight of the vehicle, then you would once again be able to show that the world's flattest tires are, indeed, flat.
According to Pascal's law, the pressure affects all directions, so I wouldn't make any distinction between horizontal and vertical pressure. The change (in hydrostatics) is only given by your $\rho gh $ formula.
Okay, more pressure should imply more temperature, but this temeprature change is completely negligible for most practical cases. Just take a book and press it against two other books (all of them at the same temperature). Then measure temperature again. You won't find any significant change, for sure.
That's because you're producing a very macroscopic force that creates a small elastic deformation. Once you stop making the force, the material liberates and gets to the same original shape. This is barely trasnferred to kinetic energy of the single molecules. You would have to somehow excite normal modes of vibration if you want to heat up the material. That's how a microwave works.
Best Answer
Yes!
The explanation is very simple . Frm first law of motion if the net force wasn't zero either the fluid or bucket would move and accelerate.