Air pressure inside pipes

The easy way to get a rough estimate is to use Bernoulli's Equation...

(P/rho + (V^2) / 2 + g*h), 1 --> 2 = 0

where P is flow pressure, rho is fluid density, V is mean flow velocity, g is the acceleration due to gravity, h is the flow height relative to some reference height, and 1 --> 2 means that you need to evaluate this sum at two points in the flow (i.e. at the pump and at the machine) and then take the difference. You can write the velocity in terms of the volume flow rate and the diameter, and then solve for diameter (everything else should be known).

Bernoulli is essentially an energy balance, and it doesn't take into account any sort of turbulence or friction (so-called turbulent drag). In addition, it strictly only applies along a flow streamline... which you won't have in turbulent flow. As your pipe is quite long, frictional head loss will probably be large and I want to emphasize that this is definitely a ROUGH estimate.

For the more exact solution, you need to deal with friction factors and Reynolds numbers. I can explain it if you're interested, but it's more of a hassle and will probably involve some numerical iteration to get an answer... let me know.

Does the orientation of the pipe matter for this solution to work? (e.g., vertical in the gravity field, or horizontal)?

No... the (g*h) term accounts for any changes in elevation relative to gravity.

You would require 1/2 inch pipe upto the pressure regulator.

Keep the length of pipe having 6psi pressure (output of regulator) as close as possible to point of use, and that will be a minimum of 3 inch.

Cheers.