May want to familiarize yourself with the USCIS issued guides. M-618 is very helpful to get started.
https://www.uscis.gov/sites/default/files/document/guides/M-618.pdf
https://www.uscis.gov/tools/how-do-i-guides/how-do-i-guides-for-permanent-residents
https://www.uscis.gov/green-card/after-we-grant-your-green-card/rights-and-responsibilities-of-a-green-card-holder-permanent-resident
I'd look for a better paying job (even if just one) so you're not saddling your family with increased cost of household - it will increase with the new immigrant arriving as she'll need to settle in and it's definitely not cheap. What are you doing for health insurance for the spouse?
If your remaining job has you comfortably over the required poverty guidelines for your household size then yes. But that doesn't sound like it is the case if you had to use your father for affidavit of support.
You could do the online Utah marriage and then go see her in Korea or she visits you in Denmark to satisfy the requirement of consummation of marriage and then file.
Get married and file for spousal visa. That one you can file online.
In the meantime sort out issues with the US: voter registration, taxes, domicile and similar as that will be needed come interview time.