You have too many restriction points in your poppet branch(s) design to allow a reasonable calculation of the actual flow rate and resulting inlet pressure loss for your assembly because there are too many restricting elements in the flow stream which includes a critical (sonic)flow determination point and multiple flow channel flow rate pressure drops including the length of the .344 dia. bore feeding your poppets, the flow restriction through the poppet spring chamber and the .015 in annular channel between the poppet stem and its surrounding bore.
However, ignoring all inlet bore, spring chamber and poppet stem .015 annulus flow losses, if you can provide me with the poppet disc lift height (is it .015 in. ?), the spring chamber bore (which is the effective poppet seat I.D.), and the O.D. of the poppet seat disc, I have access to a propriatary program that will determine the critical flow restriction point from its three possible locations and the resulting maximum air flow rate for each poppet assembly at an 80 psig inlet pressure. Also, please clarify if the .015 poppet stem to bore clearance in your drawing is radial or diametrical.
(Revision on my initally posted available analyses capability: After sending my initial post I realized that, once I determine the orfice restricted maximum flow rate, I have a another rarely used program that can calculate the pressure drop in the .344 dia x 5 in long supply port; and, therefore the maximum pressure drop you can expect in that port. However, even with this added information these calculations will not determine your actual flow through a poppet(s) because there will still be added reductions due to the potential flowing losses through the spring chamber and stem annulus regions.)
With regard to your question about lifting all four poppets at the same time, I have to ask why you are not using a single larger poppet since you specify that you are venting all of the poppets to atmosphere.