You can in all likelihood run it off your existing circuit, but without knowing the current layout it is hard to say for sure.
You are probably better off hiring an electrician. Doing work like this requires significantly more information and technique than can be explained in a forum post or single diagram.
I am a firm believer in DIY, self-reliant amateur electricians, but just reading a few posts on a forum does not cut it. You need to have and use books that explain in detail the correct procedures, otherwise you could make serious errors with potentially dangerous ramifications.
If you have a specific problem, by all means bring it to the exchange, but just asking, "How do I wire up a whole circuit?" is not a good idea.
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.
Best Answer
Short answer: NO
You may be able to use individual wires taken from a non-metallic cable (Romex is a brand) within a fixture, but not between fixtures or switches.
NEC 300.3(B) requires that:
If you give us a diagram of your fixtures and switches, we may be able to suggest a solution.