I have Likert-type data (ranked as 1 least important to 7 most important) for both dependent and independent variables. Can I use multinomial logistic regression? My second question is, if the dependent variable is binary and all independent variables are ordinal (Likert-type), would that make a difference in using a model for both type of these variables?
Ordinal logistic regression is what you need in this case. Multinomial is used when the DV is nominal. See this page for more info.
To answer your second question, logistic regression requires the dependent variable to be categorical while the independent variable(s) may be continuous or categorical, so the ordinal logistic regression will be fine.