First, the mathematical relevance would be writing down Newton's second law in vector component form. The "assumption" you mention is quite real, namely that near the surface of the earth the force of gravity acting on us is approximately constant and points down towards the surface.
Newton's second law of motion would be:
-mg = ma_y
0 = m*a_x
where y direction is vertical and x horizontal, notice that mass cancels out.
For your question Newton's second law would give...
a_y = -g
a_x = some constant
ax = f(t), some function of time.
Acceleration is the second derivative of position so either of these would give you a new equation to find x(t) and y(t). Now, since you are in high school physics I'd ask if you've have pre-calculus or calculus 1 yet. If not then going much further may not help you see how things change but I will add a little more.
What you get is a differential equation and there are techniques for solving these. You may be at the level where you have not yet learnt Newton's laws of motion but are just learning Kinematics. That's okay, we can start from that point.
a_x = dv_x/dt = d/dt(dx/dt)
a_y = dv_y/dt = d/dt(dy/dt)
These are the equations that relate acceleration to velocity and position.
The kinematics you learnt came from these equations with a_x = 0 and a_y = -g.
For your first question, what if there is a constant acceleration in the x direction, this will in fact produce a parabola but the symmetry axis will not be the vertical axis but the axis defined by the acceleration vector (a_x, a_y).
For a time dependent acceleration you need to "integrate" the second derivative twice and find the correct x(t), and y(t). That is a basic procedure.
In the real world we have air resistance that depends on the velocity of the projectile, the force is proportional to (-v_x, -v_y). The equations of motion with "linear" air resistance are solvable and you get a correction to the parabola path that involves exponential functions. You can look this up on Wikipedia, search projectiles with air resistance.