I have a data set with some categorical variables and factor scores extracted by using factor analysis. Can I use both categorical variables and factor scores in logistic regression?
Factor scores are often used in regression (including logistic regression), primarily for two reasons:
Data reduction: You have a lot of collinear variables and you want to be able to model them together. This is tricky because even if you have two variables $X_1$ and $X_2$ that are correlated (say r=0.6), they may not have the same relationship with the dependent variable. Proceed with caution.
Latent variable: You may strongly believe that the variables making up the factor scores represent an underlying latent variable/trait/construct. If that is the case, you need to do some testing, at a minimum looking at the size and direction of the factor loadings, the communality/uniqueness, and the internal consistency (say, via Cronbach's $\alpha$ or a more appropriate statistic).
Modeling a factor score, as an independent or dependent variable, is also tricky. Interpreting what a change in a factor score means can be very challenging, not only to a non-technical audience. Typically a factor score is standardized, so a 1 unit change is a change in a standard deviation - well, OK, but how do you interpret that in a practical setting? Sometimes I actually create deciles out of a factor score, which loses some information (and may lower your model fit), but can help readers interpret a result (an increase from one decile to the next may be easier to interpret than an increase in a SD). There are other options as well. I believe it is entirely context-dependent - others may disagree and argue that you go with what fits the data best, but unless you are doing prediction, I believe inferential statistics need to be interpretable to be useful and actionable (there goes my disclosure: I'm in a research shop at a government agency where I produce results for policy folks, not so much for other researchers).
One other thing to consider is measurement error. Errors-in-variables models explicitly can account for the measurement reliability of your factor scores, but I have not seen these models for logit, only continuous outcomes (I may be ignorant of existing techniques, so if others know of any please point out). You could also do latent variable modeling or structural equation modeling, where you create a path with the binary outcome measure as an observable, and the factor scores as unobservables. That can be complicated if you have never done structural equation modeling, but it may give you better results. And SEM mdoels have a ton of interesting fit statistics that are not available in standard regression models. If you go that route, just make sure that you have an estimation technique that appropriately deals with a binary outcome - you can't just take the software package's standard estimation because that is likely for a continuous outcome.
Back to your OP: can you use categorical and factor scores as independent variables? Of course. You have to take the same caution you do with any variable: how collinear are they (I hope the categorical variables are not in the factor models to produce the scores?), are they endogenous, is there measurement error, how much missing data, etc.