Would it be possible for China to be conquered by various European powers and if so when is the earliest this could happen?
The Mongols and Manchurians managed to conquer China. They had smaller numbers and more primitive technology than the Europeans. It'd definitely be an enormously difficult task and would require a large amount of luck for an appropriate opening to present itself.
The Mongols and Manchurians (not to mention the other not really Chinese states that ended up controlling China or parts of it through the millennia) had power bases right next to China, an understanding of Chinese customs, and invariable ended up somewhat Sinicized.
Sorry, I thought that there were hundreds of thousands of Russian and Japanese soldiers duking it out in Manchuria from 1904-1905. There is no power on the Planet during this time that can oppose those two in China if they work together. And why would France and Germany be opposed? Kaiser William's imperialist ambitions were well known and France would support their ally, especially if they get a slice as a reward. Of all the European Powers, only Britain would have stood up for keeping the status quo in China.
1. That was post-1900s, not pre. Better tech, more money, Trans-Siberian Railroad.
2. They both nearly went bankrupt in the process, mind you.
Plus Japan wasn't actually able to fully annex China when it tried a couple decades later, even with better technology and the Chinese divided amongst themselves (Warlords, then Central Plains, then Xinjiang, then the KMT and Communists). Though, to be fair, the Chinese did have American and British aid but that'd probably be the same here, anyways.
Neither the UK nor Germany want Russia getting a slice of China (Outer Manchuria was barely populated, not like the rest of Manchuria and China Proper), France doesn't want Germany getting anything, the US wants Open Doors, and so on.
Each one would like to annex parts of China but the status quo was still far better than letting one or two just run wild (because how would Germany and France actually defend their shares from Japan or Russia?)
China's way more valuable than Africa, both for raw resources and for captive markets, and you don't have any figures on Bismarck's level after the Iron Chancellor got sacked. It's hard to see the European Great Powers actually cooperate on something like that.
But I'd say it's not likely for it to happen pre-1900. No Trans-Siberian Railroad would make things a bit difficult for Russia, after all, seeing as they'd increase their population by 10% from taking just Manchuria and the population definitely won't be happy about having new masters. Japan's not strong enough to fight a Qing Empire in a battle of survival (Europeans thought the Qing were going to crush the Japanese in the 1890s Sino-Japanese War) and it doesn't have enough capital to have a major extended war (without foreign capital from the UK and US, who'd both oppose the war). UK opposes moves from France and Germany, Austria-Hungary is a nonfactor, US won't want it, Italy is a nonfactor probably.
Also, putting down nationalist rebellions would not be cheap.
Maybe Russia could get shafted by the other European powers in trade and completely cut out? Then, they could invade Qing near the end of its lifecycle, install a Romanov Chinese Dynasty (named something else), and forcibly put down the inevitable "anti-foreign" rebellion. Inevitably, either an organized rebel movement backed by the rest of Europe would crop up, or Europe would straight up invade if Russia blocks the rest of Europe from their puppet-China. However, if they keep trade open and flowing, especially if their emperor allows a steady, European-favorable opium-for-luxuries trade, Europe would probably support it for the most part (except for maybe Germany.)
I don't think that China would last very long after an alt-Great War or be very big, but its spin-off effects would be very interesting indeed.
Russian Chinese dynasty? Westminster would have a collective aneurism. As would Tokyo, Berlin, etc.