When I enlist for an unaccompanied assignment in Korea, how is BAH or OHA determined? Will I be able to live on base?

Practically every question you have regarding your pay and entitlements will be in DoD's "Financial Bible" (DoD 7000-14-R*, DoD's Financial Management Regulation), usually referred to as "The FMR".

I said 'practically every question' because the other aspect is determined by what the unit policies are.

If you go to South Korea on unaccompanied orders, then regardless if you're married or not, you'll be treated as a single Soldier in your living accommodations. What your actual living accommodations will be is usually according to policy and your rank. Specifically in USFK, SSG and below will live in the unit barracks.

Since you'll be assigned to the barracks, you won't be eligible for OHA for South Korea, however you will be entitled to BAH/OHA depending on where your wife lives while you are stationed overseas (suggest you read through chapter 10.4 of the FMR).

Side note: Your spouse's citizenship status doesn't matter regarding BAH/OHA. As long as the marriage is recognized by the military as a legal one and your spouse is entered into DEERS as a dependant, then the appropriate entitlements are authorized.

Additionally, suggest you start familiarizing yourself with the steps you would have to take for getting a legal status for your wife. You can start with Military One Source's webpage on Citizenship and Immigration*.

As to your last question about listing your future wife's address in the states while she lives elsewhere ... are you really soliciting advice on social media how to fraudulently claim something? The correct answer is that you list her location that she will be residing at while you are overseas and if it changes, then you are required to update the information with the military.
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* DoD Financial Management Regulation - https://comptroller.defense.gov/FMR/
* 8th Army/USFK Newcomers page - https://8tharmy.korea.army.mil/site/newcomers/
* Military OneSource (Citizenship and Immigration) - https://www.militaryonesource.mil/resources/millife-guides/citizenship-immigration-services/

I'm having a hard time believing this is a serious question but I'll play:

IF you get her info a copy of marriage cert into DEERS and IPPS-A before you leave then you'll get BAH with dependant rate. BUT, if she ends up living in her home country and NOT in the US, then your BAH rate will more than likely be based on the Zip Code of your home of record because you'll be living in the barracks of what ever base you're assigned to due to you will not be allowed to live off base by yourself....which would eliminate the need for OHA.

If you put your wife at an address in the US but they live in a different country....yea....that sounds like fraud.