The questions of naming can become very confusing. You've only touched the surface of questions that can come up. Whatever you choose to do, it is best that you try to stay as consistent as possible in your procedures. Some of this is personal preference.
1) This is an age-old question. I will usually use the foreign name for the immigrant. If that immigrant came as a young person and they have a lot of records with the Americanization, I may use the Americanization in the immigrant generation. If the next generation adopted an Americanization, I then show that spelling beginning in that generation. Different siblings may start different Americanization spellings in their respective branches, for example Fenske, Finski, Fenski, and Finske, etc.
2) I don't sync, so I can't answer your question. Do you mean that ancestry doesn't show the same "aka" name that FTM does?
3) I would enter his preferred name as Alexander Herman Fenske with a source of "Composite Name from Multiple Sources". I would enter Herman Alexander Fenske and Alex H Fenske as alternative names, with whatever sources, and Alex as an "aka" name, with its source.
4) If Elsabeth Christina Margaretha Neilson didn't come from a record, where did it come from? If that name comes from a strong source, that would be my primary name and Christina M an alternative name. I wouldn't use Christina as an "aka" as it is already in the name. We oftentimes don't know what their "real" name is (as you apparently do). It therefore is often difficult to know what the order of multiple names a person uses over their lifetime should be - especially when they use John R when younger and Robert J when older. I am usually inclined to use the name order they used later in life, but with all names spelled out (ie no initials, if I know the full name) - as the primary name.