Plumbing – Holes designed in water spigot

If it's leaking there, on a brand new hose with brand-new washer, I'm going to have to bet on the threads on the hose spigot being damaged. That should be evident if you look at them, and may be amenable to fixing with a triangular file.

The hose is not tightened, so the rubber seal is not sealing on the end of the spigot. If it's cross-threaded, or suffering thread damage, it may not be possible to tighten it until it's either put on correctly, or the damage is removed. If it's been cross-threaded with excessive force, the threads will also be damaged.

Sounds like an anti-siphon reaction to me, full water pressure in the hose going to your roof, and the house water pressure temporarily drops from the large demand from those two zones until the sprinklers are fully charged. You can try adding an anti-siphon or backflow prevention valve on the faucet to verify (they make valves that screw on between the faucet and hose). If the leaking stops or significantly reduces, then you know that's the cause.

As for why it's leaking out the stem, I'm not sure, but it's possible that's by design. The cap on top should be a vacuum breaker, allowing air in the lines after the water is shutoff and pressure drops.