I'm going to jump into this rabbit hole by relating my, ultimately positive experiences. When I bought my Long Island 1895 house, its original electrical service was 30A 110V panel with 4 15A fuses to a small number of circuits. It had been replaced with a 60A, 240V service (again with fuses) and several more circuits run, including one for an electric clothes dryer, a hot water heater, and a room air conditioner (the only circuit with Romex). Like the OP, I decided I needed more circuits, and would prefer a breaker box rather than a fuse box. I investigated having a new service run with new meter and put in a 200A breaker box myself but the restrictions were so great, I gave up on that idea. I would have had the POCO come and pull the breaker (a 2 week wait for that service), the install my new meter pan and panel, have it inspected (another 2-4 week wait), then have the POCO put in the new meter. I couldn't deal with all those delays, so I decided to go the "subpanel" route: I purchased a 40 slot load center and mounted it next to the old fuse box. I then ran 4 Ga wires from the old fuse box to the new breaker box and started moving circuits. When I had finished moving all the old circuits, I essentially abandoned the old box, using only the 60A cartridge fuses as primary protection. Finally I called a licensed electrician come in and replace the meter pan, meter, and reconnect the new panel. It took him about 2 hours to accomplish all that. While I was operating my new panel from the old 60A service, I never blew the main breaker. But that's understandable, since I had essentially no high current appliances, especially after I replaced my clothes dryer and domestic hot water heater with gas appliances. So I accomplished my original goal: more breakers to run more services, but not use significantly more power. On a side note, there was only one 15A circuit for the entire second floor; that must have been left over from the original wiring. Very difficult to run new wiring from the basement to the second floor, so I managed to run 3/4" EMT from the basement to the third floor (attic) and put in a 50A subpanel there, and run all the necessary lighting and receptacle circuits circuits from it. End of rabbit hole!

Yes, as long as you've go the load issues right, there's no problem running a sub from a sub. My house will have six panels that are effectively subpanels off the main subpanel which is distinct fro the Service disconnect.

As Randy indicated, your subpanel is missing a neutral bus bar as well as a neutral wire. You can purchase the bus bar from an electrical parts store (though I've seen them at HD/Lowes if you are using one of their panels). You will need to run a neutral wire from the neutral in the main panel to your subpanel. In the subpanel, the neutrals and grounds are to be kept separate. This is not the same as the main panel. The ground you installed at the outbuilding will be connected to the ground bar in the subpanel. You should add a red wire to your black/white pair and use it to connect to one side of the 100 amp breaker. Then move the white at both ends to neutral. That way you will have black and red as 'hots' and white as neutral.

I suspect he's out of poles in the original panel so the two circuits he's moving are ones he is getting out of the way so he can put the feeder breaker in. There's no such conduit called bendable conduit. FMC, LFMC, and FMT are restricted in their use in the Chicago code and I do not believe they can be used here. LFNC is not permitted at all. If you run RMC or EMT, you can likely use the conduit itself as the equipment ground. The THHN 12g Copper is fine for 20A even after derating. For a 50A feed, derated, you could get away with 8g Cooper. To stuff 4 #6 and 4 #12 in EMT, 3/4 is too small, 1" would be the smallest allowable size.