He wants to wire a small tiny house type mobile structure (a few outlets, lights, and one small fridge) - He is planning on using 3 circuits
He wants to connect this room To his house via standard extension cord. So his plan is to install an Eaton Load Center with 3 breakers.
So far, that's all fine. It's basically the same exact thing as hooking up an RV or travel trailer. Sticking wheels under it to make it an RV is SOP for tiny-house construction, because it dodges the anti-hovel restrictions in the building codes (350+ square feet minimum, 7 kitchen circuits, fully enclosed bathroom, etc.) But this is not a total escape; RVs have a whole chapter of the Electrical Code just for them. To escape that, you need landing gear

Or a keel
The #1 point of peril here is making the subpanel *too small*. It's super tempting to just get a $15 2-space panel and some double-stuff breakers and call it a day. I would say get at least a 16-space. You could use all of that for electric baseboard heat, dryer, A/C if you ever get a better RV style hookup.
The general classes of RV hookup are:
- 120V/15A (common plug)
- 120V/30A (TT30 plug)
- 240V/50A (NEMA 14-50 plug; standard on larger RVs).
Most campsites have all.
He then wants to wire a duplex outlet (where he can connect the extension cord to power the load center).
Because he doesn't know any better. He wants pluggability (fair enough) but the only style he knows is female on the fixed thing, so he thinks that's the only kind *there is*. And just figures we all just use male-male cords. WE DO NOT. You need to focus all your power of persuasion on this single point: NO SUICIDE CORDS.
You handle that one of two ways:
METHOD 1: Inlet. This is what he *actually wants* but does not realize exists. This bolts into a junction box like an outlet, but it actually has a backward "Plug" on it. The female end of an extension cord plugs into it, and *voilà* you are done!
However this is too expensive for some. Therefore I advise
METHOD 2: Cord-and-plug: You get a heavy extension cord 12" long and cut the socket end off of it. Take that to a proper electrical supply house and ask them for a quality strain relief for a 3/4" knockout. (that will be way too oddball for Home Depot). Then you affix a junction box to the outside of the vehicle, knock out a 3/4" hole, and stick that cord in it with the strain relief. From there, you splice to normal construction in-wall cable (UF, presumably) and onward to the panel.
For bonus points here, use a weatherproof metal box and metal conduit back to the service panel. That will make upgrades to better connectors (TT30, 14-50) easy.
Can he just connect a three wire system - one hot, one neutral and a ground from the outlet? What about grounding all of the circuits since a ground bar wasn’t included
He MUST keep ground and neutral totally separate, and remove the green bonding screw from the neutral bar. Ground bars are purchasable accessories on most panels, just a few dollars. He needs to go back and get a bigger panel anyway, so get one with an included ground bar and maybe a few breakers thrown in.
Normally I'd say get a main breaker, but if the inlet or cord end will be easy to unplug, that will suffice.