edgarh1992

Here are some really good Filipino-centric websites on Japanese visa applications.
 
https://www.thepoortraveler.net/2013/11/japan-visa-requirements-manila-philippines/
 
https://iamaileen.com/japan-visa-philippines/

Sorry I didn't get back to it.  It's been the most hectic week ever here.   
I would start with Japan.  They have more clearly defined sponsorship and somehow for us they just seemed easier.  Plus once she has a Japan visa it lends greater credibility for Korea.
 
Japan has all the paperwork online.
https://www.ph.emb-japan.go.jp/itpr_en/00_000035.html
 
These are these only travel agencies accredited by the Japanese Embassy to handle Filipino tourist visa applications.  They are who you deal with, not the embassy. 
https://www.ph.emb-japan.go.jp/itpr_ja/00_000253.html
 
Theres the application
https://www.ph.emb-japan.go.jp/files/100475146.pdf
 
Japanese example.  They are fussy.
https://www.ph.emb-japan.go.jp/files/100365623.pdf
 
Complete this letter of guarantee.
https://www.ph.emb-japan.go.jp/files/100365626.pdf
 
You need an itinerary.  Keep it simple.  You have no obligation to follow it.  Make it for a 4 or 5 day trip.  They will generally grant Filipino first time applicants a 15 day single entry visa no matter what you request.  They give longer stays and multiple entries on subsequent applications.
https://www.ph.emb-japan.go.jp/files/100365634.pdf
 
We did do a cover letter directly addressing shortcomings in my wifes application (we weren't married at the time)

How about including visa-free countries in your travel options?  For example, Singapore and Costa Rica are visa-free for Filipinos.  If your wife is not already an international traveler, adding stamps to her passport would bolster her future visa application to Japan or South Korea.
 
 
An explanation letter is not necessary, though if you want to submit one, it wouldn't hurt.  It would be clear from the visa application form that your wife is a student, she is married to a USC, you are sponsoring her, and you have a job.  As supporting evidence to the guarantor info in the form, your wife should submit to the visa agency: a copy of your marriage certificate, your recent pay stubs, and your bank statements.
 
I was unemployed when I applied for a Japanese tourist visa, so I submitted a letter with my application.  Unlike your wife, I was single, self-sponsored, and not a student.  In the letter, I explained my lack of formal employment, details about my financial capacity, and the countries I visited before.  I was granted a 30-day multiple-entry visa.
 

My wife got both Japanese and Korean visas before we were married.  It is not simple but those are both very achievable visas for Filipino passports.  I think with an American husband she will have no problems.
 
Both countries only accept visa applications through designated travel agencies in the Philippines.  You do not deal directly with the embassy and there are no interviews.  It's mainly just a big huge paperwork exercise.  Japan would generally start a a Filipino with a 15 day single entry visa.  Korea a 30 day single entry visa.  They may issue multiple entry visas and longer visits on subsequent trips.
 
When I'm in my office tomorrow I'll see if any of the paperwork we submitted would be useful as templates.  Japan wants a detailed trip itinerary etc.  
 
Also beware.  Japanese apparently really hate American sized paper. 🤣 Everything must be strictly A4 size paper.

whether you provide it or not, I expect USCIS will find it, and link it to the case 
For i-130, a U.S. passport is enough. For the visa it might be enough. 

Passport is enough for both, but I would much rather submit a one page certificate than every page of a passport.
 
Either way, the A number question and certificate number are applicable, so should be answered.  
 
Of course USCIS will find them.  Make it easy for them, by providing the information they ask for.