if domain registration is not private then they know this, if email addresses and physical addresses listed on the site are same they know same owner, if ips are the same they know same owner

if you do linking, link the lesser site to the main site only.

bottom line is if you are targeting the same keywords you are against Googles TOS because you're only supposed to have one site for your business.

If you are mostly targeting Netherlands than buy a .NL extension not DE. Secure both if expanding.

I would prefer with the amlaut if it is a German site being targeted in Germany, so it is better because people typing into google.de will be using the amlaut and this is the proper spelling for this word. That is assuming you can get a domain name with amlaut characters registered.

Will it make a big difference in rankings, probably not much so if it's a matter of getting a available domain name for cheap or paying a lot of money to purchase it off of a seller I would go with the cheaper route.

You can also look for a alternative domain name that has schädlinge with the amlaut in it and is available.

You will of course be using the amlaut in the site content and can use it in directory and filenames as well so you're still showing Google and the user what they want to see.

Use the word with the amlaut in the title tag and the meta description.

The first question I would have is why?

"The issue here is when I combine my content widget with my shop widget page, no one buys anything."

You can include all of your content on the shop page and hide it in a jquery and the shop page would look almost identical to how it is now, but that page would rank because of the content and the user would be one click closer to the purchase than if they were to land on the information page first.

I would suggest doing that and only having one page per product with the information on the page in a way that does not negatively impact conversion rates. Some a/b testing may be in order, A/B testing is always in order anyway for conversion pages

You mean in your sitemap.xml?

The canonical tag is a tab that goes on pages to let the bots know which URLs you want to get credit for content. So theoretically, your sitemap.xml file URLs should be very similar to your canonical links you are declaring in your metas.

yes noindex,follow category and tag pages. It's the Yoast default setting i believe.