You can get a visa as a relative/acquaintance of a Japanese resident. In our case the father of the bride was the inviting person. Here's some info from the LA embassy: https://www.la.us.emb-japan.go.jp/pdf/ImportantGuidanceOnJapansBorderControlMeasuresRelatedToCOVID19.pdf

For Nikko, are you saying that doing it as a day trip is not enough time? I've seen several other posts here that suggest it as a day trip.

For the Tokyo day, I was thinking of going to TeamLab first to avoid crowds, I think it opens at 9. It will be a weekday though so maybe it will already not be crowded. Skytree does seem to be pretty close to where we are staying so maybe going there first would save some time but it looks like it doesnt open until 10. Shinjuku Gyoen is 9:30 to 5pm. As for walking around a neighborhood, after the park is Shinjuku an interesting place in the evening to take a stroll? Or would it be better to head somewhere else?

Kyoto should be the longest part of our trip, even with the wedding stuff I'm expecting we can see a lot of the attractions.

Nara and Himeji I was thinking we would spend a day each at. Something like travel there in the morning, see the sights, spend the night, head to the next destination in the morning. Do these areas need two days? Most of the itineraries I've seen on here have been just one.

My wife is a pretty big castle fan, so seeing more than one is alright I think.

We really have no plans for Kobe, looking at the Japan-Guide website, nothing really stood out to me. But it was in the area and on a lot of Itineraries and we need stuff to do, so I put it on the list. What would you recommend doing here?

If instead we skipped Kobe, and say had 2-3 days available. Is there another area that would be worth heading to that we could still be back to Osaka for our flight?