GPL Licensing – Using GPL Licensed Software on Commercial Hardware

How does the use of an OS affect our own code (licence)?

GPL doesn't really stop you from distributing any software -directly or derived. So you handing over an Arch Linux CD to someone (even if you bill that person for it) - doesn't really violates anything.

Are there any effects to other components of our product, eg. the MCU part?

Secondly, if you are billing for hardware - you are only licensing for use and not for design, it is certainly fine - unless under GPLv3 you are adding restriction of usage. Hence, OS or any other software products doesn't impose restrictions on any hardware distribution. Software merely says it requires hardware of specifications or more.

Is there a possibility of avoiding GPL for our own code? eg. shipping the software seperated?

That's not quite needed I guess. However, it is important to understand that even if you ship a certain software separately, GPL (and its variants like LGPL) works on dependency. So billing separately or bundled has no difference on GPL applicability. (I will dwell more on this below.)

The node.js application delivers a WebApp to the client where it is executed. Are there any effects (As we would like to sell parts of the code as an additional App on the App Stores)?

Node.js is NOT GPL. From this: https://github.com/joyent/node/blob/master/LICENSE - but I might have overlooked something.

As long as you don't modify any software product and use it as is, you can distribute it freely (even if you are charging money to customer). If you modify existing software product and if the customer asks for the modified code - you have to handover your modification code. If your code is black box is independent of other GPL products - you can keep it closed as well.

If you feel, there is a lot of commercial value you are deriving while getting free of charge software - you might consider donating those author, but it doesn't mean you cann't use the product.