There is really no need to clarify the dynasty, like what the other posters are saying, because Chinese government from the Han to the Ming generally functioned pretty similarly despite some major administrative changes.

Or is the emperor basically a glorified job allocation manager who's main role is deciding who gets to handle doing the actual job?

The level of engagement the emperor has with the government really depends on the emperor. Some, like Emperor Wen of Sui and Yongzheng of Qing can be considered "despots" who wanted to micromanage and control everything, whereas various other emperors, like Song Huizong, preferred to pursue a life of leisure.

As an extended question, who exactly besides the emperor is doing the job of running the empire?

I made a very detailed post about this here.

Would the same have applied in that a Chinese peasant would perhaps care more about who the chief minister or who his local official/judge/administrator was, rather than what the emperor was doing?

There is a saying in Chinese: "The mountain is high and the emperor is far away" (山高皇帝遠). Especially in regions far away from the capital, local magistrates held considerable power in day to day affairs within their jurisdiction. The emperor was also removed from the general population, and the vast majority of people will never seen an emperor during his or her lifetime.

How quickly would an emperor respond to crises outside of an invasion, for example environmental disasters?

Each dynasty maintained a postal relay system. According to the Six Canons of the Tang, the Tang dynasty had 1,639 post stations empire wide which employed 20,000 people. How fast news reaches the empire depends on the level of urgency. According to Song writer Shen Kuo's Dream Pool Essays, an urgent post can go 400 li to 500 li a day. (The li unit is constantly changing, today it has been standardized to 500 meters, but during the Tang, it was around ~320 meters). Assuming that the postal relay system was working, then urgent news can reach the capital very fast. Of course, you would have to also take into account the location of the capital. Chang'an, Luoyang, Kaifeng, and Nanjing were close to the center of the empire, whereas Beijing is located far to the north.

Would he take tours of the empire periodically and basically be like, hey im the emperor everything's fine China still awesome, but don't be lazy, get back to work building that wall, make China great again, that kind of stuff?

Yes, Kangxi and Qianlong of Qing both took several "Southern Inspection Tours."

What kind of propaganda or government messages would the emperor send to his subjects?

Not sure what you mean by propaganda, but government messages includes everything related to the government from general pardons to the promulgation of new laws and regulations.

How did the empire function if the emperor was sick or on vacation? Did everything just grind to a halt, like the government is out, nobody can make official decisions, or did the government basically carry on?

The empire can run without the emperor, as we have seen in the Ming. The bureaucracy can keep running even when the emperor is absent, but this is usually because the Ming emperors employed eunuchs who acted on his behalf in dealing with the bureaucracy.

What happens if an official makes a decision that the emperor wasn't happy with, does he get executed?

Those officials are usually reprimanded and/or dismissed. Executions do happen, but it really depends. You would have to be more specific.

Was there ever a case like this where the emperor simply didn't want to run the thing and for a long period of time the empire was essentially the project of some minister or consortium of top officials?

Both the Jiajing and Wanli emperors of the Ming refused to hold court. In Jiajing's case, the bureaucracy was run by the corrupt Yan Song and a group of eunuchs, whereas during the Wanli reign, the eunuch Wei Zhongxiang ran the government.

Also what kind of careers or education would an emperor have had prior to becoming emperor?

As Crown Princes they were usually educated in the Confucian classics, in history, and post-Tang, they were required to study Tang Taizong, who was considered the model ruler.

These answers are a bit general, but if you have anything specific that you want to know, just ask and I'll try my best to explain.

You may want to clarify which dynasty you are looking for.

How quickly would an emperor respond to crises outside of an invasion, for example environmental disasters?

Before there was science to explain everything, natural disasters was a big deal for the emperor as people would interpret it as Heaven not being pleased with how the Emperor's running the Empire. If the Imperial Court does not respond immediately, it would be seen as a sign that they have lost the Mandate of Heaven and someone else with ambition would seize their chance to take this "Mandate" for themselves.

Normally, the Emperor would issue a decree condemning himself and seeking mercy from the Heavens. That usually does the trick of appeasing any rebellious thoughts. Though if these natural disasters come one after another, especially those compounded by human mismanagement and corruption, then no amount of repentance by the Emperor can save the dynasty from being overthrown.

It was said that Emperor Kangxi of the Qing Dynasty set up a fast-response task force called the Grand Council that bypasses the red tapes of the Confucian bureaucracy in order to get the latest weather and natural disaster intel before word of it spread to anyone else in the capital, so that he may prepare a response in advance. (I remember reading this in a Chinese textbook, but I can't seem to remember which book it was. Feel free to remove this post if it's not appropriate)

There is a huge difference among the descriptions you offer.

Could you be more specific in terms of the personality and time period of the Emperor?