Could Spanish ships sail up the Yalu river to intercept the Ming transports? Do they have the naval power required?
Maybe, but it might be doubtful. The ships of the era preferred to fight from a distance, and entering the Yalu would put them at greater risk, particularly from Chinese fireships and the like. They would likely have great success against the Korean and Chinese navies in the open seas, but entering rivers might be a bad move.
Anyway I doubt even the combined army can fight past the Yalu to engage Ming in China, since the Ming army, as mentioned above, still have 80% left unmobilized in case of a really serious threat, however any dragged-on wars would harm the coffers enough and contribute to the chance of Ming suing peace indeed.
Japanese troops crossed the shallower Tumen river to attack the Jurchens in Manchuria for no really good reason in OTL. They seemed to have a Korea=>Manchuria=>China trajectory going in their plans. They planned an invasion across the Yalu, but this required naval supply lines through the West Sea. In OTL, this was disrupted by the brilliant naval campaigns of Yi Sunshin, which forced the Japanese to change their plans into a simple occupation of Korea. But if the Spanish are able to nuetralise the Korean navy, then Japanese supply lines will be intact. They will likely push on for an invasion into Manchuria across the Yalu.
I think the Japanese underestimated the Jurchens OTL and would in this TL. Nurhachi, the Jurchen leader and the founder of the Qing dynasty OTL, actually offered to assist the Ming and Koreans after the Japanese attacked the Jurchens in 1592, but his offers were rebuffed (and indeed some Koreans fought with Japanese armies when they attacked the Jurchens). However, things are changed if the Japanese bring the fight into Manchuria itself. Even with complete control of the Korean peninsula, the Japanese would have to fight both the Jurchens and huge Ming armies. The better experienced and armed Japanese would have an advantage, especially with Spanish allies and solid supply lines. But Ming quantity would have a quality of its own.
Of particular importance is the fact that the Jurchens themselves could take a huge hit. This was all prior to the political, cultural and military reorganisation that created the Manchu state, the later Jin and ultimately allowed the Manchus to conquer China. This process might very well be derailed by huge hordes of samurai and Spanish mercenaries roaming around causing trouble.
As for your last point, should the nomadic presence in NE China be still as turbulent as in OTL, would any power there watch the Spanish grabbing the powerless Korean peninsula without them doing that first? The most plausible contender was the Jurchens of course.
Well, if the Japanese are able to push into Manchuria the Jurchens might take a major hit. They had not yet organised their Banner System which would prove so useful later against the Ming armies. The Jurchens themselves had enemies in the region, they may be weakened and become yet another forgotten tribe in China's northeast. This could increase the longevity of the Ming, or put them at the mercy of another invading power: the northern (and more Mongolian-influenced) Hulun Jurchens or other barbarians, or potentially a European power. I tend to think an internal rebellion is the more likely in this scenario, however.