Hello and blessings Ecclesiasticus,
I can't answer all of your questions but I hope I can provide you with insight on some of them, as we have made exactly that move. We are part of a church network focused in the Great Lakes and Upper Midwest region, centered in Wisconsin, Minnesota, Michigan and Canadian provinces (Saskatchewan, Alberta, Manitoba and Ontario) as well as two Australian provinces (Western Australia and South Australia), with thousands of us who've made the move over the years to settle and do ministry in northern Europe. It's true that the ancestral ties of our parishioners have helped in the move -- there's a big percentage of Scandinavia, German and French ancestry from these states and Canadian provinces, which has no doubt made it easier to get visas and settle in our destinations. Even though Sweden, Belgium and most of the other countries don't necessarily have official policies for automatic citizenship for their diaspora in North America and Australia, it undoubtedly does help.
Still, that's not the case for most of us-- it's generally just scattered northern European or eastern European heritage, yet we've still found ways to get the needed visas and settle, start our families and ministries in Sweden and rest of the region. Your hunch is correct that intention to start a family absolute helps to get permission to settle there, particularly if you have ancestry in those countries. After we'd gotten our initial permits and planned to get permanent visas for work and residence in Lund, we made clear in our interviews that we intended to start a family with three children. It was likely crucial for us to get the permit, since at that time we really didn't speak much Swedish, though we have since made more of an effort to learn. As far as jobs and work permits, there's a nice side benefit to the fact that we're doing ministry in our adopted lands. That is, the very act of building churches counts as skilled labor!
My husband I are not skilled IT professionals or workers with that sort of technical training. We both work in the skilled trades just as you brought up, my husband in bricklaying in masonry, me as a seamstress. Many of our parishioners in the European ministries are simply factory workers, copy editors, or service workers. But we've found a way to leverage our skills in a way that Sweden, France, Germany and the other countries find desirable, in part due to our dedication to the ministries and the skills that requires, which has helped us to gain work and residence permits there. It no doubt helps if you have technical skills, a STEM background, programming abilities, a medical or legal or other professional careers, but most of us do not. Yet we've had no trouble settling in Europe. Focus on your ministries and skills, work on efforts to convert the Syrian, Iraqi, Turkish, Kurdish and North African refugees (which are highly prized) and emphasize your intentions to start a family, and you'll have no trouble moving here. There are also enormous colonies of Americans, Canadians, Australians, New Zealanders and South Africans settled in these countries, and these networks can also be of help.