Are intrusions within a foreign embassy considered an act of war? Let's say that the U.S. breaks into the Chinese embassy in New York and decides to arrest a criminal whom the Chinese government had given permission to hole up inside the embassy, would the intrusion be considered an act of war and why?
As far as I know there also isn't an international treaty formalizing what constitutes an "act of war," so it would be up to China whether or not to use the action to justify a declaration of war on the United States. With that said, I don't believe that incursion into an embassy has been used as a justification for war, at least since the Vienna Convention on International Relations was ratified (looking at the Wikipedia page on List of Attacks on Diplomatic Missions seems to support this).
A real example would be the Iran Hostage Crisis, in which a group took over the US embassy in Tehran and held its inhabitants hostage with the eventual support of the Iranian government. In this case, while the US did not declare war on Iran it did make one attempt to rescue the hostages through military force. However, the main response of the US was to impose sanctions on Iran rather than attempt full-scale military action. If the US ever specifically called the action an "act of war," it isn't mentioned in the Wikipedia article, and wasn't used as a justification for declaring war on Iran.
On the other hand, the Chinese may decide themselves that it is an act of war. The United States has the American Service-Members' Protection Act, which allows for the President to use any means necessary, including acts of war, to bring about the release of a service member being held by the International Criminal Court. The reason I bring this up is to point out that countries get to decide for themselves what constitutes a valid reason to go to war, there really isn't anything stopping countries from going to war except other countries stepping in, and the world would have a hard time stopping a war between China and the United States through military force. Whether or not the Chinese would have a convincing argument internationally for a casus belli is going to depend on the situation and what information is available. However, if the only purpose and result of the incursion is to capture a regular criminal, most countries would probably see a declaration of war from China as an extreme overreaction and an abandonment of its international obligation to try and resolve grievances diplomatically if possible.
In May 1999, during the NATO bombing campaign in the former Yugoslavia, the USA bombed the Chinese embassy in Belgrade. Three journalists were killed and 27 other people were injured. The US said that it intended to bomb a nearby part of the Yugoslav defence ministry but a mapping error meant that the wrong coordinates were programmed into the guided bombs.
China was, of course, very angry about this but they did not go to war over the bombing. As such, it seems unlikely that a country would go to war over an intrusion by law enforcement, unless they wanted to go to war anyway and were looking for a reason.
I think there's a fundamental misconception in the question that "act of war" is anything other than a political consideration. We might discuss the Vienna Convention but that doesn't prescribe going to war over issues.
"Act of war" is whatever a country deems unacceptable and willing to fight over, and (in the modern era) what it thinks it can get UN Security Council acquiescence about. My favourite historical example is the War of Jenkins' Ear.
There is no need to dress up, wear a tie and issue a formal invitation these days.
Are intrusions within a foreign embassy considered an act of war?
In general, no this has not been the case. Unuathorised intrusions by government agencies of the hosting country are considered a diplomatic affront and a violation of international law.
would the intrusion [by US into CN embassy in US] be considered an act of war
We can only speculate, there are no rules in international law for this. I suspect it would depend on circumstances and personalities. If it happened on 21 February 1972 while Nixon and Zhou Enlai were chatting in Beijing and Nixon said it was a mistake and those committing the error would be held accountable and China recompensed then I imagine it would be quietly brushed aside by the Chinese. If it happened in 2020 while US and CN proxy forces were clashing in a resumption of the Korean war and US and CN warships colliding in disputed regions of the south china sea then I imagine it might conceivably be used as the final pretext for a formal declaration of war.
Recent history suggests that states often enter violent military conflict with one another without first formally declaring war. For example: UK-Argentina, Russia-Georgia, Russia-Crimea. The USA has not declared war on another country since 1942 but it has certainly bombed and invaded many countries since then.
This reluctance to formally declare war (e.g. in accord with 1907 Hague convention) may be due to political implications within the UN etc.
Since 1945, developments in international law such as the United Nations Charter, which prohibits both the threat and the use of force in international conflicts, have made declarations of war largely obsolete in international relations.
So notions such as "acts of war" having clear and rigid definitions may no longer be supportable. There is no internationally agreed or followed list of specific actions that constitute "acts of war". War is an informal affair these days.