Electrical – Double gang box. One switched, three hot. Is a shared neutral OK

What you will want to do in your case, instead of nutting all the neutrals together (which is technically a 310.10(H) violation!), is keep the neutral from the breaker at B1 and going to B2 and the existing light fixture separate from the neutrals coming in from B4 and going out to the two recessed-light circuits -- in other words, B1 and B5's neutrals are separate. Also, in boxes B4 and B5, keep the neutrals separated from each other as per Tester101's revised illustration -- again, NEC 310.10(H) applies.

One other tip -- the power feed to all the lights should come from the unswitched hot in B3. Otherwise, you wind up having to return the switched hots for the two four-way loops back through B4 and B5 from B3, leaving you without a neutral wire in those cables unless you switched the 14/3 for those runs out for 14/4. Using the netural from the three-way loop isn't an option because it'd create a large, annoying current loop that could interfere with the operation of timer or electronic switches that require the neutral at the switch box.

Larry, a white wire can be used as a hot and if so is supposed to be marked with black tape or otherwise marked black. You are calling it a neutral, but, if I understand your explanation, when you connect it to true neutrals and turn on the breaker it immediately trips. This means it is a hot.

Stop connecting it to neutrals. The black marked white must be feeding something. This is what I think, but I am not an experienced electrician and have never worked on anything but my own housen and it is wired in NM. Unless someone here can help you, I fully agree that it is time for you to call an electrician.

The only thing in my limited knowledge that I can imagine is that this black marked white is or was connected to a switch. This switch controlled a receptacle or half of the duplex receptacle. If the latter, the tab would have been broken and the other half powered directly. Do you have a switch near this box which seemingly doesn't do anything? Is it on? Even if the switch is off it may have been disconnected and the two wires connected together and the switch replaced. This would be contrary to code, but could have been done.

Does everything work if you just cap this black marked white wire? If so, just leave it capped and accept that you have an abandoned wire in the wall. If my suggestion is right, then one of the black wires goes to the same switch and it could also be disconnected with no loss of function.