It was the sound of the banjo in a jazz band that got me started trying to play one a year or two back (1961), and I suppose the Old Time string band music I play now has a lot in common with New Orleans jazz. There's not as much improvisation, but the swing is very similar. Old Time is definitely part of the folk scene in the US, but I get the impression it's barely tolerated over here in the UK.
As for jazz in folk clubs, I agree it's very difficult to mix the styles on the same evening, it would be like combining Indian and Chinese food in the same meal -seperately delicious but an indigestible mix.
Ray