Do colleges allow this? Is this possible…
No, you can’t major in all those things. Engineering is usually a separate faculty and you would apply to that faculty/school. There is significant overlap between biochemistry and neuroscience and you could double major in those. Some schools put computer science in with engineering and some put it in arts and science, so a double major involving computer science would depend on the school.
There are definitely universities where you can double major in an engineering field and in one or more fields of the sciences, humanities or arts. The problem with as many majors as you are contemplating is that the different major requirements might be impossible to complete in just four years of study.