Your problem is that the run from the 3 switch box to the top of the stairs doesn't have enough conductors. In addition to the hot and two switched you now have, you will need a neutral return for the new single pole section.
You could use conduit here, but there's no sense ripping out the 14/3 that's already installed. Just run a 14/2 parallel to it as shown.
I'm guessing about which switches must control which lights, but here is a suggested wiring diagram:
B = black, W = white, R = red, where indicated. Wires not marked in the diagram are all switched, so use whatever conductors are available, and please mark the ends of them with a bit of yellow tape for future maintainers. And don't forget to hook up the equipment grounding conductors!
I have omitted the details of the lights wiring as you have already done these.
Pay special attention to the connections in section 1 as the neutral wire here is carrying the load for the entire basement.
Late Edit: I just noticed that in an earlier diagram I recommended what is a code violation in most places. Justin W. has long since finished or abandoned this project, but I don't want to mislead anyone else who might read this later.
I have modified the diagram to show which conductors should be parts of a cable, in order to satisfy modern code, balance the current flow in each cable, and avoid inductive heating, magnetic radiation, and excessive line losses.
Maybe the reason J Box 1 and J Box 2 were empty is that the previous owner gave up trying to get it to work. If so, this would indicate the problem is in the cables.
If J Box 2 is in a position where a light would be installed, then the circuit was most likely originally intended to be wired like this:
(Put black tape or black paint on the ends of the white wire in the 3-wire cable to indicate that it has been re-purposed as a switched hot.)
If you wire it like this and it doesn't work, then there is probably something wrong with the 3-wire cable, Disconnect everything from the 3-wire and test all the wires for continuity. Make sure all the wire-nut connections are tight - this is where I have the most trouble in my house.
Best Answer
Untangling the smokes
What happened is that a /3 cable got run from the smoke detectors to the switch box, but with the red wire capped in the switch box as the light switch should not be connected to the interconnect wire. Then later on, when the 3-way was disconnected, the homeowner was wondering what to do with the "extra" red wire, and hooked it up to the switch as "all the wires should be hooked up", in his mind at least.
However, that's not how it works. With the breaker off, take that red wire out of the switch and cap it off with a wirenut once again (you may need to tape the nut on to get it to hold). Now, you won't have to worry about any issues that could be caused with your smoke detectors by the light-switching arrangement.
What you'll need to do is run a switch spur from the existing switch to the new switch
Once you have the situation with the smoke detectors sorted, run your /3 cable between the old and new switch boxes. The ground wire in the new cable ties to a grounding screw in the old box (since it appears to use screws on a backstrap for grounding instead of the normal scheme) and to the grounding arrangement in the new box, while the black wire from the existing /3 cable nuts to the white wire (taped black) in the new /3 cable instead of being run into the switch, and the red and black wires in the new /3 cable connect to the brass screws (use the screws please!) on the existing 3-way switch. While you're here, move the black wire from the /2 cable from the backstab to the black screw terminal on the switch.
In the new switch box, tape the white wire black and connect it to the black screw on the new 3-way switch. The red and black wires then land on the brass screws of the new 3-way. Once you're done with both switch boxes, then you can button everything up, turn the breaker back on, and enjoy!