How to reuse parts of WordPress site e.g. header, footer, part of header for multiple WordPress sites?

I would think that your best bet is option 3 (multisite) if possible. Once you’ve got your network of sites set up you can then specify which menu to use as described here.

Doing the other solutions (a more rough and ready ‘hardcoded’ approach) will almost certainly give you headaches in the future. Having multisite set up also has advantages such as ease of updating and maintenance of user base.

You’re right that if you use a multisite setup, all sites use one database. You shouldn’t lose any data in the handover, but you should definitely do a complete backup first in case. If you’re worried, do a dry run by setting up a testing environment and exporting / importing the data using Tools > Export / Import.