Wiring – connect a water heater to an existing dual-pole breaker

There can only be one point in the system where neutral and ground wires are joined, and this is usually in the main panel. Therefore, you must lift the bond from the generator, or disconnect the ground wire from the generator to the transfer switch.

If you lift the bond on the generator, you can run both ground and an insulated neutral back to the transfer switch. You will also run ground and insulated neutral from the main panel to the transfer switch, and ground and insulated neutral to the subpanel from the transfer switch. Note that neither neutral nor ground are switched in the transfer switch. The generator can optionally be grounded to a rod.

If you cannot lift the bond, disconnect the ground wire to the transfer switch at the generator (or don't run a ground wire at all). The subpanel will still be grounded from the unswitched ground wire from the main panel. The generator cannot be grounded to a rod because this would tie the neutral to ground in a second location through the bond.

You can do it -- you'll need a subpanel loadcenter in a NEMA 3R enclosure, though, as well as watertight conduit/raceway and fittings for the outdoor runs, which will be done using THWN in the conduits.

You will want to put a 30A 2pole for the dryer and a 20A GFCI for the laundry outlets, as well as a spare 20A breaker (if you want a lighting circuit there, or an extra laundry outlet circuit) in your subpanel, which does not need a main breaker of its own, so it can be a four-slot NEMA 3R loadcenter, such as an Eaton BR48L125RP with a GBK10 ground bar in it, as the neutral and ground will be on separate, isolated bars in the subpanel.

As to the feeder breaker, the NEC calls for 1500VA for the laundry outlet circuit (220.52(B)) and a minimum of 5000VA (a 20A dryer) or the wattage of the dryer (whichever is higher) for an electric dryer load (220.54). Assuming your dryer pulls the full 30A (7500VA) and you have 1500VA of laundry circuit load, this yields 9000VA or 37.5A, which requires a 40A feeder breaker in the main panel, and 8AWG THWN for the feeder wires. In reality, your dryer can't sit there and pull 30A continuously without tripping your breaker, so this is a conservative calculation.