If you are going for gnat's bottom accuracy, I'd suggest a single stage. Mostly because that's what I and most everybody else uses for loading rounds that exceed the quality and consistency of any factory match rounds.
Most progressives are capable of producing very high quality ammo, but my opinion is that they simply aren't capable of producing as repeatable results as a single stage is capable of - more moving parts means less chance of true repeatability. That is not to say that a LnL, Dillon or RCBS cannot produce very high quality ammo, because they can, but I wouldn't use one for any of my hunting ammo or if I was shooting in a match with a good rifle.
Concerning using a lubing die, the LnL has 5 independent stations - you can use them for whatever you want. There is no rule that says station 1 has to be size/deprime.
RE: neck sizing, there are different schools of thought on that one. I used to neck size, but don't do it anymore. Use good lube and a good sizing die and you don't overwork your brass unless you have a larger than normal chamber in your rifle. But we could start 27 different threads concerning the opinions on neck versus full length versus small base versus shoulder bump sizing.
For better than factory plinking or hunting ammo, a progressive is the ticket. For shooting 1 hole groups at whatever range you are looking for, I'd advise to use a single stage.