Is there some mathematical derivation for calculating the orbital velocity based on altitude and acceleration without using calculus? I thought of equations of motion, but I always get problems.
Is there a way to derive it using laws of gravitational potential and kinetic energy ?

Are the pieces of information here in the video correct ? I guess it has mistaken with the position of the satellite if not affected with gravity.

You actually just need very little calculus, but you can instead think of it as the limit approaching to zero rather than the derivative and that should help you.

Actually even the notion of instantaneous speed is based on calculus. So if you use instantaneous speed in the formula for acceleration you "use calculus".
Defining it as a limit rather than a derivative is still calculus. The notion of limit is one of the basic notions in calculus. And the derivative itself is such a limit.

So technically, you cannot do kinematics at all without calculus. You may have the illusion that you do without. :)
After all, Newton had to invent it before he can study motion.

Even if you want to use geometry, you will have to use it in a way that allows for calculus (i.e. allow for derivatives) ...otherwise words like "Acceleration" or "velocity" won't make any sense...
As for basic algebra, I don't know what that would mean... like addition, multiplication etc?