Yes, you definitely make use of research questions and hypotheses simultaneously! There are many journal articles, theses other research output forms that include both!
The answer is definitely yes! In many cases, you can have quantitative research questions that can be converted to hypotheses. Other research questions must be dealt with using qualitative methods. When you have both qualitative and quantitative procedures in the same study, it is referred to as a mixed methods study. There is a large literature base available on mixed methods.
For the qualitative part of your research, a lot depends on its role within the overall mixed methods research design. In general, hypotheses are rare in qualitative research, but they may be appropriate in a mixed methods design. So, you need to tell us more about your research and the connection between your methods.
It depends on the type of case study research. If it is a case study-explanatory sequential mixed methods design, you can use research question(s) and hypothesis/hypotheses together in a study. For instance, Alotumi (2022) applied research questions and hypotheses in his case study-mixed methods research, in which qualitative findings clarified the quantitative results of his hypothesized relationships. Here is the full citation.
Alotumi, M. (2022). Factors influencing graduate students’ behavioral intention to use Google Classroom: Case study-mixed methods research. Education and Information Technologies. https://doi.org/10.1007/s10639-022-11051-2
Yes, Mr. Bade, you can have both research questions and hypotheses at the same time. But you have to be careful in the case when you are going for exploratory research, not explanatory. You can not have a hypothesis in exploratory research, where you don't know the independent and dependent variables to test the hypothesis.
Traditionally, but that can be changing, hypotheses are used in quantitative experimental and descriptive designs. Usually research questions are used in correlational studies.
Research questions have traditionally been used with qualitative designs. The results of a qualitative study are often presented as hypothesis that should be further investigated. Historically qualitative studies have small number of subjects leading to poor opportunity for quantitative analysis of the results.