Am I oblidged to give a copy of the license to the end customer? If yes, in what form? Written paper delivered together with the device, on the LCD display during FW runup, ... ?
Yes. Section 4 states that when you distribute a work that is licensed under the Apache License 2.0, you must give the recipients a copy of the license. It doesn't matter how you distribute the licensed work (as source files, a compiled binary) or if you have made modifications to the Apache-licensed source code. You also need to carry along any copyright, patent, trademark, and attribution notices from the license and make those visible.
However, the license does not specify the format of providing the license. For example, looking at some Android apps, they have a "credits" screen that allows the user to see the various open source components they use, the author name(s), and licenses. Some desktop applications do this, too. If your product doesn't have user interactions, you could include this information in the box on a user's manual or flash it to an output screen on boot.
The closest thing that you'll find to guidance is Section 4.d. If the source form has a NOTICE file, you need to carry along the information in the NOTICE file. That may be in a NOTICE file as well, or it could be "whereever such third-party notices normally appear".
I would have to inform the end-customer about changes I did on the original software files? The customer doesn't get the changed files, just the final product, including object code.
Your source files need to have a notice that the original files were licensed under Apache License 2.0 and have been modified by you (or, since it's likely that your company holds the copyright to the modifications, by your company). If you aren't distributing your source files, then your customers will never see this notice, and that's OK. It's for people who have source code access to understand that parts of the software are Apache licensed.