How important is it to correct Landsat for atmosphere, haze and convert DN to reflectance? What impact does it have on the results, say for many of the vegetation indices?
Generally the answer is "Quite a large effect", but it can vary significantly. In general - do atmospheric correction if possible, and the best atmospheric correction you can - although I appreciate this isn't always easy.
As for a few more details:
Contributions from the atmosphere to NDVI are significant (McDonald et al., 1998) and can amount to 50% or more over thin or broken vegetation cover (Verstraete, 1994)
(Quote from Song et al., 2001 - available at http://www.unc.edu/courses/2008spring/geog/577/001/www/Song01_RSE.pdf (The Verstraete source is a book which isn't freely available)
In fact, the whole of the Song (2001) reference is a very good answer to your question about when it is useful to do atmospheric correction, and when you needn't do it. It also answers the related question about which method (may) be best to use.
There are cpf files available. There is more information here. http://landsathandbook.gsfc.nasa.gov/cpf/
Not a real answer but a great reference nonetheless.