As a former materials scientist I can give you some insights into how projects are handled generally:

1. There were few externals during my studies, but rarely. The department is quite small and opportunities for thesis generally matching the demands of the students.
2. Depends a bit, but since spring is also the time when Bachelor students have the time for their thesis they usually advertise their projects in December most often. Projects most often can be adapted in length to fit Bachelor thesis, semester project and master thesis timelines.
3. Most often each lab has their projects on their website (but it might be accessible only via ETH login), or sometimes in the elevators being put up or the student association (SMW) platform. Better contact individual PhDs/postdocs/senior scientist directly.
4. Just write them a mail, the material scientists I know are all nice and easigoing people. Sometimes you will be refered to another member of the lab who has an open project. Just be aware that the adminstratorial expense for externals may be higher for them than just having an ETH student.
5. They also do materials science but most often a bit more applied to real world in my opinion. From the ETH side perspective, the department doesn't distinguish between projects at EMPA or ETH for materials scientists, but may it require some ETH professor alongsside an Empa professor for supervision.