I don't think Russia decides to test NATO's article 5 should they win in Ukr.
W/ regards to Taiwan long term, I would expect China to try and co-opt through legal means as they did in HK, but that failing they may take a more aggressive route.
We've got TSMC plants being built in the US and the designs of the chip architecture occur here. So we've got the intellectual property, just need to add the manufacturing capacity. We also have other chip fabs being built. One of the few things I think Biden did correctly was the CHIPS act though I'm sure there are parts of it I would detest if I went through the whole text of it.
If I was in China's shoes and had committed to taking Taiwan (which no one can know whether they actually plan to follow through on at this point) I would plan or try to do so before Ukr is concluded. You don't want to try to take w/ America's undivided attention on you.
As such I don't think the fall of Ukr would be some sort of go signal for them. Sure it would be one more symbol of the crumbling Global American Empire, but strategically the conclusion of the UKR conflict can only hurt them with regards to Taiwan.
Should the US and China go at it, I don't see WWIII happening as you envision, China's nuclear capacity is such that it loses any exchange that should occur. I do see what could be a very costly conventional war for the US. The Sad fact is that their anti ship missiles are cheaper than our boats, and should they manufacture enough of those missiles they can send enough of them so that they saturate our missile defense systems and some missiles get through. Given the range of their IRBM's the US has already had to develop the capacity for a stealth drone to refuel our planes just to get them to the battle space given how far away the Carriers would need to position themselves to limit the threat from Chinese anti ship missiles.
To that effect here's the boeing MQ-25 stealth drone refueling platform we've developed.
Our newer carriers now have the capacity to launch lighter aircraft like this b/c of the development of electromagnetic launch systems (replacing steam versions) on our newer carriers.
If they manage to sink one of our carriers that's 6000 sailors dead (two 9/11's for those keeping track).
Given all this I'm not confident we could win a conflict with china over Taiwan. I'm not confident we would lose either, but the fact that I don't know either way is troublesome.
Now we could embargo the strait of Malacca keeping any fuel from the Middle East from getting to China and shut them down in about 6 months (russia's oil export capacity isn't such that it could fill the gap over land through siberia) but I don't think the US could stomach the global consequences of the reaction to deciding that free trade isn't free on the high seas anymore. More to the point the US and Chinese economy are still so deeply entangled that I don't think we would be willing to suffer the domestic consequences of shutting down China.
Now should we get to a point where we can survive domestically without the current high level of Chinese trade, that calculus may change. But as it stands now I don't think either country can afford for a taiwan altercation to grow beyond a regional conventional conflict.