For 2D projectile motion, why is the velocity in the horizontal direction constant?
I learned that we neglect air resistance, but why?
And isn't there horizontal acceleration at the beginning when an object is launched?
The "projectile motion" that you study when starting dynamics is a simplified version of "real life" which is makes the mathematics easy. By assuming the acceleration due to gravity is constant and ignoring any other forces on the body, you don't even need calculus to set up and solve simplified versions of problems that are at approximations to "real life" scenarios.
Air resistance does not act in the horizontal direction. It acts in the direction of motion of the projectile, so it affects the vertical motion as well as the horizontal. The magnitude of the air resistance force depends on the speed of the projectile, but there is no simple "equation" relating the force to the speed.
Those facts mean that to calculate what happens "in real life," in most situations you can't even find a "mathematical formula" for the answer using calculus - you have to use numerical methods and a computer. Learning how to do that is not the objective of the "projectiles" section of a first dynamics course.
To answer your other question, if you think about it, for the problems you are solving the projectile is not really "launched" at all. It just happens to have a given position and velocity at time zero. What it was doing before time zero is irrelevant to the question.
But you are right that in reality, to launch a projectile from rest you do need to apply a force to it. The magnitude of the force you apply depends on the time it takes to launch the projectile, which in real life can not be "instantaneous". The shorter the time, the bigger the force, but Newton's second law means that the product "average force $\times$ time" is what determines the projectile velocity immediately after it is launched.
So you can simplify the mathematics (again!) by assuming the launch is instantaneous, and finding the amount of "force $\times$ time" which starts the motion. In fact that quantity has its own name, "impulse", and Newton's second law can be stated as "impulse = change of momentum," where momentum is defined to be mass $\times$ velocity.