If you believe that your Likert item DV is tapping a continuous underlying quantity, eg. a strength of preference for or against something (which I presume you do), then your regression model should assume an ordinal DV because only then will you by trying to estimate the quantities you want: the effects of changes in IVs on the actual satisfaction level underlying the DV.
If for some reason you doubt that the DV is really measuring something continuous then you might fit a multinomial regression to see whether particular combinations of IVs predict that people choose particular categories of the DV.
You might also fit a multinomial on a collapsed set of DV categories if respondents don't use the whole range or if you don't have enough data to get an ordinal model going. An ordinal model has to estimate the boundaries between DV categories as well as the slopes for the IVs, so it can demand more from your data.
The type and distribution of the IVs makes no difference since no regression models make any distributional assumptions about them.