It is possible with a relevant BA degree to enroll in a Master’s program. If your program is academically high enough and you had courses related to ID you definitely have a chance to enroll. You have to check the specific requirements of the program, usually under ‘Admission’ on their website. You will often be assessed individually on eligibility and you will need a good grade average as well.
I think that depends on both the school and the student, and what your goals are. Some masters programs are true masters in that they focus much more on theory. So if you goal is to switch specializations and you are looking for the skills you need to do that, some masters programs would not be the right fit for that.
If you have a clear vision for yourself as a student, plus the presentation materials to show that you can manage to reach your vision within given time and circumstances, you will have a good chance to succeed. Especially for programs having an overlap with ID, and I’d say interior design has a 50-60% overlap in skills and work process, it will be possible to make the transfer. If the mindset of a designer is there already, you will have to work extra hard to master the new skills and theoretical background, but it will make it possible.