You can do factor analysis on dichotomous and ordinal measures; some care needs to be taken, but it can be done.
However, I don't think that is the best way to reduce the size of the questionnaire. I would focus on pairs of questions, using measures of association that are appropriate to the different scales. Then I'd look for patterns that are very common across more than one question. You can also look at correlations between each question and the whole scale (however you are calculating it). You might even try cluster analysis.
If your MCQ are "choose one" then you can look at the choices for each pair of questions in 2x2 tables. If there is a pair where nearly all of the answers are in one cell, then one of the questions is a candidate for deletion.
For two Likert scales you can use rank correlation - pairs with corr close to 1 yield a candidate for deletion.
Then you can look for combinations of 3 items; one way to do this is to list each combination of items and see if any dominate.
This takes longer than factor analysis, but I think it gives you a better feel for what is going on.