1. Baekje-Nippon Union
I thought this already happened in OTL from 500 to the mid 600s, especially considering the Battle of Baekgang, but I'm assuming you want a more permanent union. In order to accomplish this, one possibility, though unlikely, is Goguryeo absorbing Silla in a similar manner as they did with Buyeo in the mid or late 400s, forcing Baekje to consolidate its hold on Chinese colonies and tighten its relation with Japan. In this case, Goguryeo would dominate on land, while Baekje would have to aggressively pursue naval domination in order to maintain its trading posts/colonies. However, this also assumes that King Jangsu would have no ambition of unifying the peninsula. If he found a reason to absorb Silla, then it is also likely that he would have viewed Baekje as a threat.
Another possibility is to weaken Silla enough so that Baekje experiences a resurgence in the 500s, eventually managing to dominate the peninsula. When the Sui or the Tang invades Goguryeo, Baekje could work with them, or could aid Goguryeo but force it to concede to certain terms, weakening the northern state. However, this is tricky as well because Chinese reunification will ultimately mean conflict with Baekje's colonies on the coast. Because we have a lack of records attesting what happened in OTL, it's hard to say whether Goguryeo could have taken advantage by invading peninsular Baekje, or the Sui and Baekje could have worked out terms through diplomacy.
I'm assuming you mean the state established in 918, in which case the rulers' titles already indicated that they considered themselves as emperors until the Mongols/Yuan lowered their status. However, in order to increase their power, they could ironically take a diplomatic approach. Balhae had fallen to the Liao only 8 years earlier, and King Taejo (Wang Geon) had welcomed the crown prince along with numerous refugees. Because Balhae most likely had a court historian as well, Goryeo could have asked for a map and some documents proving their territorial hold on Manchuria, and negotiate with the Liao. Of course, the Khitan could have also invaded outright, causing more problems, but considering how Balhae remnants resisted foreign rule for about 200 years, and how Goryeo actively resisted the Liao, it is possible that had they joined forces, Goryeo might have gained a stable hold on southern Manchuria.
I already addressed a few possibilities here, although in the last possibility, I meant the 1800s and not the 1900s. I would also like to add that Joseon became submissive to the Ming and the Qing and adapted neo-Confucianism mostly due to invasions from the Yuan, Japanese, and Manchus (Qing) over a period of 400 years. There were occasional lulls as well, but the chaos was devastating enough.
As a side note, looking at your maps, the analogues in my timeline for the territories of Goryeo and Joseon are similar to that of yours, even though my POD will occur in the 400s, with a possible earlier one in the 200s BC.