To be considered as in the US you have to be on the sponsoring employee's official orders. As you're not, the time away without approved re-entry permit can be problematic. If your spouse is going to remain outside the US you'll definitely want to go back to the US and apply for the re-entry permit properly (or even stay in the US the necessary time so you don't lose your green card).
You can get married in Germany or even with the Utah online marriage. Timeline won't change - it's about 18 months to immigrant visa, depending on the backlog of people waiting for visa interviews in your consulate. You won't get a green card until after you've entered the US and activated your immigrant visa (and paid the green card issuance fee to USCIS).
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Lots of red flags. Large age difference. Unauthorized work. Why is it unauthorized? Is she on an actual student visa (F1) or did she do a change of status from tourist visa to attend college and can't leave country without having to go to Bogota and get a real F1 issued?
Oh and USCIS does take fraud marriages seriously so think twice before going down that path.
What are you confused about? Make an account on IRS.gov, download your tax return transcripts, print them and bring them to your interview. Took me a total of under 15 minutes to get mine the other week.
With this set of circumstances you should be able to only have mom's income qualify for I-134/I-864 but they could come back to you and ask for dad's income too as he's a member of household.
For starters, don't try to claim child as your own as that will not end well. So she shouldn't list your last name on BC. Get the father unknown BC and then K-2.
She'll have to figure out the passport for the child - hire a local lawyer knowledgeable in these things - I'm sure the lawyer can research the local laws for parentage and passport issues.
There will be nothing you need to provide for her prior stay in the US - just the Japan police/court certificate - whatever the requirements are for that consulate.