Electronic – Power connectors for very long LED strips

The typical rule of thumb for these strips is injecting power every 5m or better. Power should be injected at both ends, for the best color matching. FPC, flexible printed copper has relatively high resistance and it quickly adds up.

You need 1890W ( (15m * 7) * 3.6A ) * 5v. You mentioned the supply's power efficiency at 80%, but that should already be accounted for in it's stated output capacity. On the other hand, a 20% margin of safety, as to not drive the supplies at their maximum output, is a good idea. 2300W it is.

You have two options. Get a few 5v high current supplies, and run some cables in parallel to each 4~5m section. Due to the high current, this can get pretty expensive, as each 5m section takes 18 Amps. A quick calculation with a AWG calculator, At a 32 foot run (5 Meters to and from, you have to double the cable run) with 18A at 6V, you get a 0.6V drop... USING 10 AWG. That's 10% wasted in pretty thick and expensive cable. You could go with a 12v supply instead, at 18A, 10M, 20AWG, would result in 50% voltage drop, giving you 6v at the target end. Still not efficient.

The other option, is to get some high voltage, low current supplies (Say 48v), and some small dc-dc buck switchers able to handle 18A output. You only need 90W at the target. At 48V, 90 Watts is a little under 2 amps. 2.5 Amps, to adjust for the cable drop and switcher efficiency. This way you only need some 22AWG cable, resulting in a 2.5% voltage drop, which doesn't effect the switcher at all. You could even run it all in series, as the resistance drop at 22 AWG and 48v can be very negligible with the right switchers. And you only need a 48v 50A (2400W) supply to power it all. Way better than 5v 480A and high gauge cables.

Since you are using the same power supply for both, the Arduino and the LEDs, you do not need to connect the ground pins -- as long as ground is connected internally.

As you have recognized this would be redundant and unnecessary and could even lead to further problems (there would be several ground loops). So I think connecting the Arduino to ground just one time will be fine.