Are you talking more of how the region of modern China has been fragmented, or what historians considered to be "China" as a whole? There's a difference because quite a few fragmented dynasties occupied regions which are no longer part of modern China.

There were several times when China was fragmented into multiple dynasties.

• The Three Kingdoms, 三国 - after the Han dynasty, due mainly to the Yellow Turban Rebellion (黄巾之乱). This period ended with Cao Wei leaving the strongest, however Sima Yi then took over control over Cao Wei and conquered the Shu, eventually resulting in his grandson founding the Jin Dynasty. The Three Kingdoms lasted for ~60 years

• The 16 States, 十六国 - split up northern China during the collapse of the Jin dynasty. This was between non-Han ethnic groups that had lived in the northern regions of China. Political upheaval during the Jin dynasty weakened it's control over the northern regions, and several independent rebellions led to the creation of the 16 states. It ended when the Northern Wei unified northern China. This period lasted for ~150 years,

• The North and South Dynasties, 南北朝 - during the unification of northern China, an East Jin commander Liu Yu (劉裕) took control of the throne and established the Liu Song dynasty. It ended when a commander, Yang Jian (楊堅), under the Chen dynasty (occupying approximately the same area as the Liu Song) took control and established the Sui dynasty, and then proceeded to unify north and south again. This lasted for ~170 years.

• The Five Dynasties and Ten States, 五代十国 - during the slow decline of the Tang dynasty, two large-scale rebellions, the An Shi Rebellion (安史之乱) and the Huang Chao Rebellion (黄巢之乱) severely impacted the central government's power.

  The An Shi Rebellion started when Commander An Lushan (安禄山) declared himself emperor of Northern China, and ended after several years of war. The Huang Chao Rebellion, which happened approximately a century later and lasted for a decade, was more of a civilian rebellion, where common people rose up against the oppressive government.

  A few years after the second rebellion the Tang dynasty fell and disintegrated, with various military leaders establishing their independent states in the south, or dynasties in the north. During this time, the Liao dynasty in the north consolidated its power, eventually taking control over the entire northern China (most of modern Mongolia, northern Korea, eastern regions of Russia, and north-eastern China). In the south, the ten states eventually fell under control of the Song. Ended with the establishment of the Song dynasty in the south (who eventually managed to unify north and south China a century later). Lasted for ~75 years.

• A small separation during the Song dynasty between the Jin (金), West Xia (西夏), and the Song (宋) during the Mongol invasions of Genghis Khan. Ended with the establishment of the Yuan Dynasty under Kublai Khan.

So, to do the math:

Total division time (and a few other small, insignificant periods) = ~500 years

Imperial China (Qin unification to Qing disintegration) = ~2100 years

~~Ancient and Imperial China (Xia to Qing) = ~4000 years~~

So to answer your second question, no, much less than half of its existence.

Sources:

The Rise of Modern China - Hsu, I.

China: A New History - Fairbank, J. K.

Let me show you the timeline of China.

1.1 west Zhou. The west Zhou's capital is in Shaanxi province.

1.2.1 east Zhou, Spring and Autumn period. After an invasion of west nomadic people, Quanrong(not the later famous Xiongnu), the Zhou royal authority had to move east to Henan province. And this gave the King's principalities a signal that the King, Son of Heaven, was just so so. During his exodus, the Qin tribe (Till then, Qin was not a principality, because Qin's ancestors had offended Zhou's founders.) escorted the King. Thanks for Qin's doing, the King gave them a rank of nobility, viscount-the lowest in rank, but it entitled Qin a qualification to play game of throne. Also, the King gave Qin the land where Zhou's founders initially thrived.

1.2.2 east Zhou, warring states period.

2 Qin dynasty.

3.1 west Han dynasty.

3.2 east Han dynasty.

4 three kingdoms period.

5.1 west Jin dynasty(Sima family)

5.2.1 east Jin dynasty, with five minorities (Xiongnu, Xianbei, Qiang, Jie, and Di) invading north China.

5.2.1 east Jin dynasty, with Di's Qin in north China.

6.1 four consecutive dynasties: Song dynasty of Liu family (not the later Song of Zhao family), Qi dynasty of Xiao family, Liang dynasty of Xiao family(This Xiao was the former Xiao's remote relative.), and Chen dynasty of Chen family(Yes, the dynasty name was the same as family name.). And with the Xianbei's north Wei dynasty in north China.

6.2 three kingdoms period-Chen, Sui, and Qi.

to be continued...