antss, you're making the rather broad assumption that all of us contributing to these forums even live in the same country, let along the same neighbourhood! I have no idea where you live, but in my Canadian city, I need separate circuits for all major appliances (even my microwave shelf required a separate circuit for its plug), make up air for all venting over 250cfm, and kitchen ventilation is definitely required for all new builds and renovations. And you'd better remove your shoes before you enter any private home here! (That's not code, just polite practice.) I only mention this to reinforce the idea that one should not come to a forum like this to get building code information. It varies too widely, and we don't know where anyone lives unless they choose to tell us. Check your local ordinances, or even a reliable contracting company. Then install your appliances in accordance with the manufacturer's recommendations if you want to maintain your warranties, plus your home insurance coverage.

You have a neutral wire loose, probably due to a push-in connection at a receptacle. I would check the last working receptacle (closest to a non working one) on the circuit for the loose wire.

Gray area. Since it didn't do this from the day it was installed, it is unlikely that it is actually wired wrong. Therefore, a component has failed (I'm guessing top heating element as shorted internally). It's two screws, two wires, and a big nut. Either is qualified to find the problem and replace it, but the plumber is more likely to already have a replacement in stock, so I personally would aim this at the plumber. Re-reading your original post, it's 7 months old, there's probably a warranty, so call back whoever installed it (most likely a plumber).

I take it you got this motor used? I think you've probably analyzed the situation pretty well. It does appear that the starting winding failed (it reads open when measured for resistance), and that some previous owner tried hacking the motor, hoping to make it work without the starting winding. The fact that it hums and then will run in either direction if you hand start it sounds about right. I'm far from a motor expert, but I think that capacitor start motors also use a starting winding. Without a working one in your motor, I think you're probably stuck. But I will defer to anyone else with better motor chops.