To become a teacher, you only need to have two key subjects - Higher English and National 5 Maths. You do need additional Highers, which should be from a wide range of different subjects. However, if you are looking to become a particular type of teachers, e.g. Maths Teacher, you would be expected to take your Higher/Advanced Higher Mathematics as well, which will allow you entry into a Maths-based course.
If a course asks for AABB, that would mean that they can be in (almost) any subject. If the course requires specific courses to be passed, at specific grades then it would specify on the course page.
For engineering, it can really depend on the type of engineering you are looking to study. This is because there are differences between the types of engineering you can study - some may be more chemistry focused whereas some may be more physics focused.
In general, the following subjects could be required for engineering:
Not necessarily! For most courses, they would look for you to have taken 2-3 different science courses at Higher. However, most universities would count Mathematics as a higher science - which can count as one of the 3. Generally, they would expected you to have taken both Chemistry and Biology.
Not necessarily! For some courses, they might have the school subject as an entry requirement, but a lot of more general courses, (for example) business, would not require you to take Higher Business. Alongside this, not every school will offer this subject!